text: SXDF-ALMA deep survey 3 Figure 1. (left) ALMA 1.1 mm continuum image obtained using a 19-point mosaic (indicated by white circles). Detected 23 ALMA sources are indicated by small yellow circles (larger circles denote a higher S/N) with their source ID, and the distribution of Hα emitters at z = 2.53 is indicated by small magenta circles. Small green circles denote the MIPS 24µm sources. (middle) A magniﬁed view of the ALMA image showing two bright sources (corresponding to SXD- F-ALMA1 and 2 in Table 1). (right) A 3-color composite NIR image of the same region. There are 3 Hα emitters, 3 MIPS sources (two of these are overlapped), and 1 star-forming galaxy at z = 1.382. We note a variety of 1.1 mm properties among these galaxies. spec Table 1. A catalogue of ALMA 1.1 mm continuum sources1. ID α δ Speak S/N weight2 note3 ± ± SXDF-ALMA1 02 17 40.524 -05 13 10.64 1.696 ± 0.058 29.36 0.952 Hα emitter z = 2.53 ± 0.02 SXDF-ALMA2 02 17 41.120 -05 13 15.19 0.791 ± 0.065 14.27 0.852 Hα emitter z = 2.53 0.02 SXDF-ALMA3 02 17 43.642 -05 14 23.81 0.839 0.090 9.29 0.609 Herschel/JVLA dropout ± SXDF-ALMA4 02 17 42.335 -05 14 05.09 0.395 ± 0.056 7.05 0.982 zphoto = 1.33+−00..1106 SXDF-ALMA5 02 17 41.229 -05 14 02.74 0.378 ± 0.056 6.70 0.976 zphoto = 1.52+−00..1138 SXDF-ALMA6 02 17 41.597 -05 13 12.29 0.315 ± 0.067 4.73 0.827 SXDF-ALMA7 02 17 41.153 -05 12 45.47 0.299 ± 0.065 4.58 0.844 SXDF-ALMA8 02 17 39.678 -05 13 22.81 0.282 ± 0.062 4.54 0.884 SXDF-ALMA9 02 17 41.270 -05 14 01.62 0.259 ± 0.057 4.52 0.960 SXDF-ALMA10 02 17 42.965 -05 13 51.46 0.270 0.061 4.44 0.905 1Only the sources with S/N larger than 4.4 are shown. 2primary beam correction factor. In this table, Speak has been corrected accordingly (1/weight). 3source properties taken from Yamaguchi et al. (in prep.) tections, SXDF-ALMA3, is extremely faint even in the ultra-deep NIR images from CANDELS and HUGS, as shown in Figure 2. Further contributions to the star forma- tion history may come from these faint submm galaxies that do not appear to be fully overlapped with UV/optical-selected galaxies (e.g., Chen et al. 2014). Another new type of ALMA sources is the mm-wave line-emitting galaxy; we ﬁnd a ∼ promising candidate for a line-emitting galaxy at an observing frequency of 273.3 GHz ± ± with a peak ﬂux of 3.8 0.70 mJy (5.4σ) or a velocity-integrated line ﬂux of 0.53 0.079 − ∼ − Jy km s 1 (6.7σ) that exhibits a galaxy-like line width (FWHM 100 km s 1). Although it is not yet clear at this stage whether this is a [CII] emitter at z = 5.955 or a low-z CO emitting galaxy, this result encourages us to search for such mm-wave line emitters using the ALMA data. This may allow us to constrain [CII] and/or CO luminosity functions across the history of the universe (e.g., Ono et al. 2014, Tamura et al. 2014, Matsuda et al. 2015). Implications for future ALMA deep surveys: We ﬁnd a rapid increase in the number ∼ of faint ALMA sources below 0.3 mJy as expected by the latest source counts around ∼ 1 mm, demonstrating that undertaking a shallower survey by a factor of 2 would dras- ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SXDF-ALMA

text: deep

text: survey

text: Figure

text: (left)

text: ALMA

text: mm

text: continuum

text: image

text: obtained

text: using

text: a

text: mosaic

text: (indicated

text: by

text: white

text: circles).

text: Detected

text: ALMA

text: sources

text: are

text: indicated

text: by

text: small

text: yellow

text: circles

text: (larger

text: circles

text: denote

text: a

text: higher

text: S/N)

text: with

text: their

text: source

text: ID,

text: and

text: the

text: distribution

text: of

text: Hα

text: emitters

text: at

text: z

text: is

text: indicated

text: by

text: small

text: magenta

text: circles.

text: Small

text: green

text: circles

text: denote

text: the

text: MIPS

text: sources.

text: (middle)

text: A

text: magniﬁed

text: view

text: of

text: the

text: ALMA

text: image

text: showing

text: two

text: bright

text: sources

text: (corresponding

text: to

text: SXD-

text: and

text: in

text: Table

text: (right)

text: A

text: composite

text: NIR

text: image

text: of

text: the

text: same

text: region.

text: There

text: are

text: Hα

text: emitters,

text: MIPS

text: sources

text: (two

text: of

text: these

text: are

text: overlapped),

text: and

text: star-forming

text: galaxy

text: at

text: z

text: We

text: note

text: a

text: variety

text: of

text: mm

text: properties

text: among

text: these

text: galaxies.

text: spec

text: Table

text: A

text: catalogue

text: of

text: ALMA

text: mm

text: continuum

text: ID

text: α

text: δ

text: Speak

text: S/N

text: Hα

text: emitter

text: z

text: Hα

text: emitter

text: z

text: Herschel/JVLA

text: dropout

text: zphoto

text: zphoto

text: the

text: sources

text: with

text: S/N

text: larger

text: than

text: are

text: shown.

text: beam

text: correction

text: factor.

text: In

text: this

text: table,

text: Speak

text: has

text: been

text: corrected

text: accordingly

text: properties

text: taken

text: from

text: Yamaguchi

text: et

text: al.

text: (in

text: prep.)

text: tections,

text: is

text: extremely

text: faint

text: even

text: in

text: the

text: ultra-deep

text: NIR

text: images

text: from

text: CANDELS

text: and

text: HUGS,

text: as

text: shown

text: in

text: Figure

text: Further

text: contributions

text: to

text: the

text: star

text: forma-

text: tion

text: history

text: may

text: come

text: from

text: these

text: faint

text: submm

text: galaxies

text: that

text: do

text: not

text: appear

text: to

text: be

text: fully

text: overlapped

text: with

text: UV/optical-selected

text: galaxies

text: (e.g.,

text: Chen

text: et

text: al.

text: Another

text: new

text: type

text: of

text: ALMA

text: sources

text: is

text: the

text: mm-wave

text: line-emitting

text: galaxy;

text: we

text: ﬁnd

text: a

text: promising

text: candidate

text: for

text: a

text: line-emitting

text: galaxy

text: at

text: an

text: observing

text: frequency

text: of

text: GHz

text: with

text: a

text: peak

text: ﬂux

text: of

text: mJy

text: or

text: a

text: velocity-integrated

text: line

text: ﬂux

text: of

text: Jy

text: km

text: s

text: that

text: exhibits

text: a

text: galaxy-like

text: line

text: width

text: (FWHM

text: km

text: s

text: Although

text: it

text: is

text: not

text: yet

text: clear

text: at

text: this

text: stage

text: whether

text: this

text: is

text: a

text: [CII]

text: emitter

text: at

text: z

text: or

text: a

text: low-z

text: CO

text: emitting

text: galaxy,

text: this

text: result

text: encourages

text: us

text: to

text: search

text: for

text: such

text: mm-wave

text: line

text: emitters

text: using

text: the

text: ALMA

text: data.

text: This

text: may

text: allow

text: us

text: to

text: constrain

text: [CII]

text: and/or

text: CO

text: luminosity

text: functions

text: across

text: the

text: history

text: of

text: the

text: universe

text: (e.g.,

text: Ono

text: et

text: al.

text: Tamura

text: et

text: al.

text: Matsuda

text: et

text: al.

text: Implications

text: for

text: future

text: ALMA

text: deep

text: surveys:

text: We

text: ﬁnd

text: a

text: rapid

text: increase

text: in

text: the

text: number

text: of

text: faint

text: ALMA

text: sources

text: below

text: mJy

text: as

text: expected

text: by

text: the

text: latest

text: source

text: counts

text: around

text: mm,

text: demonstrating

text: that

text: undertaking

text: a

text: shallower

text: survey

text: by

text: a

text: factor

text: of

text: would

text: dras-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: -σ 5 xy Support for this research at the University of Wisconsin- Madison was provided by the Oﬃce of the Vice Chancel- 3 lor for Research and Graduate Education with funding 2 from the Wisconsin Alumni Research Foundation. Sup- port for this research at Michigan State University was 1 ϵ provided by the Institute for Mathematical and Theo- 1 retical Physics with funding from the oﬃce of the Vice 2 President for Research and Graduate Studies. - 5 0 5 10 15 m FIG. 4: Comparison of the AHE in the cases of a smooth [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 disorder potential (dotted) and point like scatterers (solid). (2010). We assumed m = m > 0 and additionally imposed v = v in Γ Γ [2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 the case of short ranged impurities. For the blue curves, we − (2011). set (cid:15) = (cid:15) while the red curves are obtained for (cid:15) = (cid:15) 7m. Γ Γ [3] B. Bernevig and T. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, 2013), ISBN 9780691151755. states and impurities. Indeed, the analysis of various lim- [4] M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. iting cases of the three-band Dirac model reveals that the Rev. Lett. 104, 106408 (2010). large energy asymptote of the anomalous Hall response [5] M. Dzero, J. Xia, V. Galitski, and P. Coleman, preprint | | scales as (m/ (cid:15) )3 in the case of smooth impurity poten- arXiv:1506.05635 (2015). ∼ | | [6] F. Lu, J. Zhao, H. Weng, Z. Fang, and X. Dai, Phys. tial while σ m/ (cid:15) for short range scatterers. This xy Rev. Lett. 110, 096401 (2013). behavior persists in the generic result. In contrast, in the ∼ [7] V. Alexandrov, M. Dzero, and P. Coleman, Phys. Rev. Bychkov-Rashba model σ m/(cid:15)2 [31]. xy Lett. 111, 226403 (2013). As mentioned in the introduction, present day experi- [8] M. Ye, J. Allen, and K. Sun, preprint arXiv:1307.7191 mental samples are believed to host a multitude of large (2013). [9] B. Roy, J. D. Sau, M. Dzero, and V. Galitski, Phys. Rev. ferromagnetic domains. In our theory, smooth ﬂuctua- B 90, 155314 (2014). tions of the magnetization can be taken into account by [10] A. Menth, E. Buehler, and T. H. Geballe, Phys. Rev. averaging the ﬁnal result. Even after this procedure, the Lett. 22, 295 (1969). asymptotics allow to distinguish smooth and sharp im- [11] J. W. Allen, B. Batlogg, and P. Wachter, Phys. Rev. B purity potentials in the described manner. Up to now 20, 4807 (1979). magnetotransport experiments on TKIs concentrated a [12] S. Wolgast, C. Kurdak, K. Sun, J. W. Allen, D.-J. Kim, hysteretic behavior in the longitudinal conductance. Sys- and Z. Fisk, Phys. Rev. B 88, 180405 (2013). [13] D. J. Kim, S. Thomas, T. Grant, J. Botimer, Z. Fisk, tematic investigation of the transverse conductance is and J. Xia, Scientiﬁc Rep. 3, 3150 EP (2013). still needed. [14] S. Thomas, D. Kim, S. Chung, T. Grant, Z. Fisk, and In this paper we analyzed the semiclassical AHE and X. J. (2013), arXiv1307.4133. uncovered the importance of diﬀractive skew scattering [15] M. Neupane, N. Alidoust, S. Xu, T. Kondo, Y. Ishida, in the context of topological Kondo insulators. In a par- D.-J. Kim, C. Liu, I. Belopolski, Y. Jo, T.-R. Chang, allel vein, our results have further rich consequences for et al., Nat. Comm. 4 (2013). [16] N. Xu, X. Shi, P. K. Biswas, C. E. Matt, R. S. Dhaka, anomalous transport phenomena in other multiband ma- Y. Huang, N. C. Plumb, M. Radovi´c, J. H. Dil, E. Pom- terial systems such as Weyl semimetals [42] and chiral jakushina, et al., Phys. Rev. B 88, 121102 (2013). p-wave superconductors [43–45]. Quantum eﬀects, such [17] J. Jiang, S. Li, T. Zhang, Z. Sun, F. Chen, Z. Ye, M. Xu, as interaction and localization corrections to the conduc- Q. Ge, S. Tan, X. Niu, et al., Nat. Comm. 4 (2013). tivity tensor, the quantum AHE [46–48] and the surface [18] G. Li, Z. Xiang, F. Yu, T. Asaba, B. Lawson, P. Cai, state quantum Hall eﬀect [37, 49–51] on TKIs remain a C. Tinsman, A. Berkley, S. Wolgast, Y. S. Eo, et al., theoretical and experimental challenge for the future. Science 346, 1208 (2014). [19] S. Wolgast, Y. S. Eo, T. O¨ztu¨rk, G. Li, Z. Xiang, C. Tins- Acknowledgements. We thank A. Andreev, I. man, T. Asaba, B. Lawson, F. Yu, J. W. Allen, K. Sun, Dmitriev, M. Khodas, L. Li, J. Sauls, K. Sun and M. L. Li, C¸. Kurdak, D.-J. Kim, and Z. Fisk, Phys. Rev. B Titov for important discussions. P.M.O. and E.J.K. ac- 92, 115110 (2015). knowledge hospitality by the Department of Physics and [20] Y. Nakajima, P. S. Syers, X. Wang, R. Wang, and Astronomy at Michigan State University, and by the De- J. Paglione, preprint arXiv:1312.6132 (2013). partment of Physics at University of Michigan (E.J.K.). [21] Z.-H. Zhu, A. Nicolaou, G. Levy, N. P. Butch, P. Syers, X. F. Wang, J. Paglione, G. A. Sawatzky, I. S. Elﬁmov, This work was ﬁnancially supported in part by NSF and A. Damascelli Phys. Rev. Lett. 111, 216402 (2013). Grant No. DMR-1506547 (M.D.), and NSF Grants No. [22] P. Hlawenka, K. Siemensmeyer, E. Weschke, DMR-1606517 and ECCS-1560732 (E.J.K. and A.L.). ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: -σ

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text: oﬃce

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text: President

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text: and

text: Graduate

text: Studies.

text: m

text: FIG.

text: Comparison

text: of

text: the

text: AHE

text: in

text: the

text: cases

text: of

text: a

text: smooth

text: M.

text: Z.

text: Hasan

text: and

text: C.

text: L.

text: Kane,

text: Rev.

text: Mod.

text: Phys.

text: disorder

text: potential

text: (dotted)

text: and

text: point

text: like

text: scatterers

text: (solid).

text: We

text: assumed

text: m

text: m

text: and

text: additionally

text: imposed

text: v

text: v

text: in

text: Γ

text: Γ

text: X.-L.

text: Qi

text: and

text: S.-C.

text: Zhang,

text: Rev.

text: Mod.

text: Phys.

text: the

text: case

text: of

text: short

text: ranged

text: impurities.

text: For

text: the

text: blue

text: curves,

text: we

text: set

text: while

text: the

text: red

text: curves

text: are

text: obtained

text: for

text: Γ

text: Γ

text: B.

text: Bernevig

text: and

text: T.

text: Hughes,

text: Topological

text: Insulators

text: and

text: Topological

text: Superconductors

text: (Princeton

text: University

text: Press,

text: ISBN

text: states

text: and

text: impurities.

text: Indeed,

text: the

text: analysis

text: of

text: various

text: lim-

text: M.

text: Dzero,

text: K.

text: Sun,

text: V.

text: Galitski,

text: and

text: P.

text: Coleman,

text: Phys.

text: iting

text: cases

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text: Lett.

text: large

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text: anomalous

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text: response

text: M.

text: Dzero,

text: J.

text: Xia,

text: V.

text: Galitski,

text: and

text: P.

text: Coleman,

text: preprint

text: scales

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text: (m/

text: in

text: the

text: case

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text: smooth

text: impurity

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text: F.

text: Lu,

text: J.

text: Zhao,

text: H.

text: Weng,

text: Z.

text: Fang,

text: and

text: X.

text: Dai,

text: Phys.

text: tial

text: while

text: σ

text: m/

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text: This

text: xy

text: Rev.

text: Lett.

text: behavior

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text: generic

text: result.

text: In

text: contrast,

text: in

text: the

text: V.

text: Alexandrov,

text: M.

text: Dzero,

text: and

text: P.

text: Coleman,

text: Phys.

text: Rev.

text: Bychkov-Rashba

text: model

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text: xy

text: Lett.

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text: M.

text: Ye,

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text: Allen,

text: and

text: K.

text: Sun,

text: preprint

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text: large

text: B.

text: Roy,

text: J.

text: D.

text: Sau,

text: M.

text: Dzero,

text: and

text: V.

text: Galitski,

text: Phys.

text: Rev.

text: ferromagnetic

text: domains.

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text: B

text: tions

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text: be

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text: account

text: by

text: A.

text: Menth,

text: E.

text: Buehler,

text: and

text: T.

text: H.

text: Geballe,

text: Phys.

text: Rev.

text: averaging

text: the

text: ﬁnal

text: result.

text: Even

text: after

text: this

text: procedure,

text: the

text: Lett.

text: asymptotics

text: allow

text: to

text: distinguish

text: smooth

text: and

text: sharp

text: im-

text: J.

text: W.

text: Allen,

text: B.

text: Batlogg,

text: and

text: P.

text: Wachter,

text: Phys.

text: Rev.

text: B

text: purity

text: potentials

text: in

text: the

text: described

text: manner.

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text: now

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text: a

text: S.

text: Wolgast,

text: C.

text: Kurdak,

text: K.

text: Sun,

text: J.

text: W.

text: Allen,

text: D.-J.

text: Kim,

text: hysteretic

text: behavior

text: in

text: the

text: longitudinal

text: conductance.

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text: and

text: Z.

text: Fisk,

text: Phys.

text: Rev.

text: B

text: D.

text: J.

text: Kim,

text: S.

text: Thomas,

text: T.

text: Grant,

text: J.

text: Botimer,

text: Z.

text: Fisk,

text: tematic

text: investigation

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text: J.

text: Xia,

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text: Rep.

text: EP

text: still

text: needed.

text: S.

text: Thomas,

text: D.

text: Kim,

text: S.

text: Chung,

text: T.

text: Grant,

text: Z.

text: Fisk,

text: and

text: In

text: this

text: paper

text: we

text: analyzed

text: the

text: semiclassical

text: AHE

text: and

text: X.

text: J.

text: uncovered

text: the

text: importance

text: of

text: diﬀractive

text: skew

text: scattering

text: M.

text: Neupane,

text: N.

text: Alidoust,

text: S.

text: Xu,

text: T.

text: Kondo,

text: Y.

text: Ishida,

text: in

text: the

text: context

text: of

text: topological

text: Kondo

text: insulators.

text: In

text: a

text: par-

text: D.-J.

text: Kim,

text: C.

text: Liu,

text: I.

text: Belopolski,

text: Y.

text: Jo,

text: T.-R.

text: Chang,

text: allel

text: vein,

text: our

text: results

text: have

text: further

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text: for

text: et

text: al.,

text: Nat.

text: Comm.

text: N.

text: Xu,

text: X.

text: Shi,

text: P.

text: K.

text: Biswas,

text: C.

text: E.

text: Matt,

text: R.

text: S.

text: Dhaka,

text: anomalous

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text: phenomena

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text: other

text: multiband

text: ma-

text: Y.

text: Huang,

text: N.

text: C.

text: Plumb,

text: M.

text: Radovi´c,

text: J.

text: H.

text: Dil,

text: E.

text: Pom-

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text: systems

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text: semimetals

text: and

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text: et

text: al.,

text: Phys.

text: Rev.

text: B

text: p-wave

text: superconductors

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text: eﬀects,

text: such

text: J.

text: Jiang,

text: S.

text: Li,

text: T.

text: Zhang,

text: Z.

text: Sun,

text: F.

text: Chen,

text: Z.

text: Ye,

text: M.

text: Xu,

text: as

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text: Q.

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text: S.

text: Tan,

text: X.

text: Niu,

text: et

text: al.,

text: Nat.

text: Comm.

text: tivity

text: tensor,

text: the

text: quantum

text: AHE

text: and

text: the

text: surface

text: G.

text: Li,

text: Z.

text: Xiang,

text: F.

text: Yu,

text: T.

text: Asaba,

text: B.

text: Lawson,

text: P.

text: Cai,

text: state

text: quantum

text: Hall

text: eﬀect

text: on

text: TKIs

text: remain

text: a

text: C.

text: Tinsman,

text: A.

text: Berkley,

text: S.

text: Wolgast,

text: Y.

text: S.

text: Eo,

text: et

text: al.,

text: theoretical

text: and

text: experimental

text: challenge

text: for

text: the

text: future.

text: Science

text: S.

text: Wolgast,

text: Y.

text: S.

text: Eo,

text: T.

text: O¨ztu¨rk,

text: G.

text: Li,

text: Z.

text: Xiang,

text: C.

text: Tins-

text: Acknowledgements.

text: We

text: thank

text: A.

text: Andreev,

text: I.

text: man,

text: T.

text: Asaba,

text: B.

text: Lawson,

text: F.

text: Yu,

text: J.

text: W.

text: Allen,

text: K.

text: Sun,

text: Dmitriev,

text: M.

text: Khodas,

text: L.

text: Li,

text: J.

text: Sauls,

text: K.

text: Sun

text: and

text: M.

text: L.

text: Li,

text: C¸.

text: Kurdak,

text: D.-J.

text: Kim,

text: and

text: Z.

text: Fisk,

text: Phys.

text: Rev.

text: B

text: Titov

text: for

text: important

text: discussions.

text: P.M.O.

text: and

text: E.J.K.

text: ac-

text: knowledge

text: hospitality

text: by

text: the

text: Department

text: of

text: Physics

text: and

text: Y.

text: Nakajima,

text: P.

text: S.

text: Syers,

text: X.

text: Wang,

text: R.

text: Wang,

text: and

text: Astronomy

text: at

text: Michigan

text: State

text: University,

text: and

text: by

text: the

text: De-

text: J.

text: Paglione,

text: preprint

text: partment

text: of

text: Physics

text: at

text: University

text: of

text: Michigan

text: (E.J.K.).

text: Z.-H.

text: Zhu,

text: A.

text: Nicolaou,

text: G.

text: Levy,

text: N.

text: P.

text: Butch,

text: P.

text: Syers,

text: X.

text: F.

text: Wang,

text: J.

text: Paglione,

text: G.

text: A.

text: Sawatzky,

text: I.

text: S.

text: Elﬁmov,

text: This

text: work

text: was

text: ﬁnancially

text: supported

text: in

text: part

text: by

text: NSF

text: and

text: A.

text: Damascelli

text: Phys.

text: Rev.

text: Lett.

text: Grant

text: No.

text: (M.D.),

text: and

text: NSF

text: Grants

text: No.

text: P.

text: Hlawenka,

text: K.

text: Siemensmeyer,

text: E.

text: Weschke,

text: and

text: (E.J.K.

text: and

text: A.L.).

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 A. Varykhalov, J. Sa´nchez-Barriga, N. Y. Shitse- valova, A. V. Dukhnenko, V. B. Filipov, S. Gaba´ni, K. Flachbart, O. Rader, E. D. L. Rienks, preprint arXiv:1502.01542 (2015). [23] N. Nagaosa, J. Sinova, S. Onoda, A. MacDonald, and N. Ong, Rev. Mod. Phys. 82, 1539 (2010). [24] N. A. Sinitsyn, J. of Phys.: Cond. Matt. 20, 023201 (2008). [25] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954). [26] J. Smit, Physica 21, 877 (1955). [27] L. Berger, Physica 30, 1141 (1964). [28] P. Streda, J. Phys. C 15, L717 (1970). [29] N. Sinitsyn, A. MacDonald, T. Jungwirth, V. Dugaev, and J. Sinova, Phys. Rev. B 75, 045315 (2007). [30] I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, Europhys. Lett. 111, 37004 (2015). [31] I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, preprint arXiv:1511.07413 (2015). [32] A. A. Abrikosov, L.P. Gor’kov, and I. E. Dzyaloshin- skii, Quantum Field Theoretical Methods in Statistical Physics, (Pergamon, 1965). [33] N. A. Sinitsyn, J. E. Hill, H. Min, J. Sinova, and A. H. MacDonald, Phys. Rev. Lett. 97, 106804 (2006). [34] Maximally crossed diagrams can form an exception, since they lead to weak localization corrections that are en- hanced logarithmically. [35] Supplementary materials to this article. [36] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976). [37] E. Ko¨nig, P. Ostrovsky, I. Protopopov, I. Gornyi, I. Bur- mistrov, and A. Mirlin, Phys. Rev. B 90, 165435 (2014). [38] Ming-Che Chang and Qian Niu, Phys. Rev. Lett. 75, 1348 (1995). [39] A. A. Kovalev, Y. Tserkovnyak, K. Vy`borny`, and J. Sinova, Phys. Rev. B 79, 195129 (2009). [40] A. A. Kovalev, J. Sinova, and Y. Tserkovnyak, Phys. Rev. Lett. 105, 036601 (2010). [41] M. Dzero, M. G. Vavilov, K. Kechedzhi, and V. Galitski, Phys. Rev. B 92, 165415 (2015). [42] A. A. Burkov, Phys. Rev. Lett. 113, 187202 (2014). [43] Roman M. Lutchyn, Pavel Nagornykh, Victor M. Yakovenko, Phys. Rev. B 80, 104508 (2009). [44] E. Tailor, C. Kallin, Phys. Rev. Lett. 108, 157001 (2012). [45] Songci Li, A. V. Andreev, B. Z. Spivak, Phys. Rev. B 92, 100506 (2015). [46] H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, Adv. in Phys. 64, 227 (2015). [47] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, preprint arXiv:1508.07106 (2015). [48] C.-Z. Chang and M. Li, preprint arXiv:1510.01754. [49] X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science 323, 1184 (2009). [50] C. Bru¨ne, C. X. Liu, E. G. Novik, E. M. Hankiewicz, H. Buhmann, Y. L. Chen, X. L. Qi, Z. X. Shen, S. C. Zhang, and L. W. Molenkamp, Phys. Rev. Lett. 106, 126803 (2011). [51] R. Yoshimi, A. Tsukazaki, Y. Kozuka, J. Falson, K. S. Takahashi, J. G. Checkelsky, N. Nagaosa, M. Kawasaki, and Y. Tokura, Nat. Comm. 6 (2015).

text: A.

text: Varykhalov,

text: J.

text: Sa´nchez-Barriga,

text: N.

text: Y.

text: Shitse-

text: valova,

text: A.

text: V.

text: Dukhnenko,

text: V.

text: B.

text: Filipov,

text: S.

text: Gaba´ni,

text: K.

text: Flachbart,

text: O.

text: Rader,

text: E.

text: D.

text: L.

text: Rienks,

text: preprint

text: N.

text: Nagaosa,

text: J.

text: Sinova,

text: S.

text: Onoda,

text: A.

text: MacDonald,

text: and

text: N.

text: Ong,

text: Rev.

text: Mod.

text: Phys.

text: N.

text: A.

text: Sinitsyn,

text: J.

text: of

text: Phys.:

text: Cond.

text: Matt.

text: R.

text: Karplus

text: and

text: J.

text: M.

text: Luttinger,

text: Phys.

text: Rev.

text: J.

text: Smit,

text: Physica

text: L.

text: Berger,

text: Physica

text: P.

text: Streda,

text: J.

text: Phys.

text: C

text: N.

text: Sinitsyn,

text: A.

text: MacDonald,

text: T.

text: Jungwirth,

text: V.

text: Dugaev,

text: and

text: J.

text: Sinova,

text: Phys.

text: Rev.

text: B

text: I.

text: A.

text: Ado,

text: I.

text: A.

text: Dmitriev,

text: P.

text: M.

text: Ostrovsky,

text: and

text: M.

text: Titov,

text: Europhys.

text: Lett.

text: I.

text: A.

text: Ado,

text: I.

text: A.

text: Dmitriev,

text: P.

text: M.

text: Ostrovsky,

text: and

text: M.

text: Titov,

text: preprint

text: A.

text: A.

text: Abrikosov,

text: L.P.

text: Gor’kov,

text: and

text: I.

text: E.

text: Dzyaloshin-

text: skii,

text: Quantum

text: Field

text: Theoretical

text: Methods

text: in

text: Statistical

text: Physics,

text: (Pergamon,

text: N.

text: A.

text: Sinitsyn,

text: J.

text: E.

text: Hill,

text: H.

text: Min,

text: J.

text: Sinova,

text: and

text: A.

text: H.

text: MacDonald,

text: Phys.

text: Rev.

text: Lett.

text: Maximally

text: crossed

text: diagrams

text: can

text: form

text: an

text: exception,

text: since

text: they

text: lead

text: to

text: weak

text: localization

text: corrections

text: that

text: are

text: en-

text: hanced

text: logarithmically.

text: Supplementary

text: materials

text: to

text: this

text: article.

text: R.

text: Jackiw

text: and

text: C.

text: Rebbi,

text: Phys.

text: Rev.

text: D

text: E.

text: Ko¨nig,

text: P.

text: Ostrovsky,

text: I.

text: Protopopov,

text: I.

text: Gornyi,

text: I.

text: Bur-

text: mistrov,

text: and

text: A.

text: Mirlin,

text: Phys.

text: Rev.

text: B

text: Ming-Che

text: Chang

text: and

text: Qian

text: Niu,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: A.

text: Kovalev,

text: Y.

text: Tserkovnyak,

text: K.

text: Vy`borny`,

text: and

text: J.

text: Sinova,

text: Phys.

text: Rev.

text: B

text: A.

text: A.

text: Kovalev,

text: J.

text: Sinova,

text: and

text: Y.

text: Tserkovnyak,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Dzero,

text: M.

text: G.

text: Vavilov,

text: K.

text: Kechedzhi,

text: and

text: V.

text: Galitski,

text: Phys.

text: Rev.

text: B

text: A.

text: A.

text: Burkov,

text: Phys.

text: Rev.

text: Lett.

text: Roman

text: M.

text: Lutchyn,

text: Pavel

text: Nagornykh,

text: Victor

text: M.

text: Yakovenko,

text: Phys.

text: Rev.

text: B

text: E.

text: Tailor,

text: C.

text: Kallin,

text: Phys.

text: Rev.

text: Lett.

text: Songci

text: Li,

text: A.

text: V.

text: Andreev,

text: B.

text: Z.

text: Spivak,

text: Phys.

text: Rev.

text: B

text: H.

text: Weng,

text: R.

text: Yu,

text: X.

text: Hu,

text: X.

text: Dai,

text: and

text: Z.

text: Fang,

text: Adv.

text: in

text: Phys.

text: C.-X.

text: Liu,

text: S.-C.

text: Zhang,

text: and

text: X.-L.

text: Qi,

text: preprint

text: C.-Z.

text: Chang

text: and

text: M.

text: Li,

text: preprint

text: X.-L.

text: Qi,

text: R.

text: Li,

text: J.

text: Zang,

text: and

text: S.-C.

text: Zhang,

text: Science

text: C.

text: Bru¨ne,

text: C.

text: X.

text: Liu,

text: E.

text: G.

text: Novik,

text: E.

text: M.

text: Hankiewicz,

text: H.

text: Buhmann,

text: Y.

text: L.

text: Chen,

text: X.

text: L.

text: Qi,

text: Z.

text: X.

text: Shen,

text: S.

text: C.

text: Zhang,

text: and

text: L.

text: W.

text: Molenkamp,

text: Phys.

text: Rev.

text: Lett.

text: R.

text: Yoshimi,

text: A.

text: Tsukazaki,

text: Y.

text: Kozuka,

text: J.

text: Falson,

text: K.

text: S.

text: Takahashi,

text: J.

text: G.

text: Checkelsky,

text: N.

text: Nagaosa,

text: M.

text: Kawasaki,

text: and

text: Y.

text: Tokura,

text: Nat.

text: Comm.


text: Theorem 17 (Natural sparse closure of cycles). Consider a packing integer program as deﬁned J ⊆ in (P). Let 2[n] be a partition of the index set of columns of A and let GpacJk be the packing A, interaction graph of A. If GpacJk is a cycle of length K, then: A, ∈ Z ≤ 1. If K = 3k, k , then zN.S. 3zI. ++ 2 ∈ Z ≤ 2. If K = 3k + 1, k , then zN.S. 3k+1zI. ++ 2k ∈ Z ≤ 3. If K = 3k + 2, k , then zN.S. 3k+2zI. ++ 2k+1 V Moreover, for any (cid:15) > 0, there exists a packing integer program with a suitable partition of (cid:0) (cid:1) variables, where GpacJk is a cycle of length K such that A, ∈ Z ≥ −(cid:0) (cid:1) 1. If K = 3k, k , then zN.S. 3 (cid:15) zI. ++ 2 (cid:16) (cid:17) ∈ Z ≥ − 2. If K = 3k + 1, k , then zN.S. 3k+1 (cid:15) zI. ++ 2k ∈ Z ≥ − 3. If K = 3k + 2, k , then zN.S. 3k+2 (cid:15) zI. ++ 2k+1 All proofs of the above results are presented in Section 2.1. 2.3 Covering problems In this section, we present our results on the quality of sparse cutting-planes for covering- type problems, that is problems of the following form: (C) min cT x ≥ s.t. Ax b ∈ Z ∀ ∈ L x , j j + ∈ R ∀ ∈ \L x , j [n] j + ∈ Q × ∈ Q ∈ Q L ⊆ with A m n, b m, c n and [n]. In this case, we would like to prove lower + + + bounds on the objective functions after adding the sparse cutting-planes. Our ﬁrst observation is a negative result: super sparse cuts as deﬁned for the packing-type problems can be arbitrarily bad for the case of covering problems. In order to present this result, let formalize the notion of super sparse cuts in this setting. In particular, given an J { } instance of type (C), we assume we partition the variable indices n into = J , J , . . . , J . ∈ 1 2 q For all j [q] we add all possible cuts that have support on variables with index in J . Let j zS.S. be the optimal objective function of the resulting LP with cuts. Theorem 18. For any constant M > 0, there exists a covering integer program (C) and J { } partition := J , J of [n], such that the corresponding zS.S. and zI satisﬁes: 1 2 · zI > M zS.S.. 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Theorem

text: (Natural

text: sparse

text: closure

text: of

text: cycles).

text: Consider

text: a

text: packing

text: integer

text: program

text: as

text: deﬁned

text: J

text: in

text: (P).

text: Let

text: be

text: a

text: partition

text: of

text: the

text: index

text: set

text: of

text: columns

text: of

text: A

text: and

text: let

text: GpacJk

text: be

text: the

text: packing

text: A,

text: interaction

text: graph

text: of

text: A.

text: If

text: GpacJk

text: is

text: a

text: cycle

text: of

text: length

text: K,

text: then:

text: A,

label: list

text: Z

text: If

text: K

text: k

text: then

text: zN.S.

text: Z

text: If

text: K

text: k

text: then

text: zN.S.

text: Z

text: If

text: K

text: k

text: then

text: zN.S.

text: V

text: Moreover,

text: for

text: any

text: there

text: exists

text: a

text: packing

text: integer

text: program

text: with

text: a

text: suitable

text: partition

text: of

text: variables,

text: where

text: GpacJk

text: is

text: a

text: cycle

text: of

text: length

text: K

text: such

text: that

text: A,

text: Z

text: If

text: K

text: k

text: then

text: zN.S.

text: zI.

text: Z

text: If

text: K

text: k

text: then

text: zN.S.

text: zI.

text: Z

text: If

text: K

text: k

text: then

text: zN.S.

text: zI.

text: All

text: proofs

text: of

text: the

text: above

text: results

text: are

text: presented

text: in

text: Section

text: Covering

text: problems

text: In

text: this

text: section,

text: we

text: present

text: our

text: results

text: on

text: the

text: quality

text: of

text: sparse

text: cutting-planes

text: for

text: covering-

text: type

text: problems,

text: that

text: is

text: problems

text: of

text: the

text: following

text: form:

text: (C)

text: min

text: cT

text: x

text: s.t.

text: Ax

text: b

text: Z

text: L

text: x

text: j

text: j

text: R

text: \L

text: x

text: j

text: [n]

text: j

text: Q

text: Q

text: Q

text: L

text: with

text: A

text: m

text: n,

text: b

text: m,

text: c

text: n

text: and

text: [n].

text: In

text: this

text: case,

text: we

text: would

text: like

text: to

text: prove

text: lower

text: bounds

text: on

text: the

text: objective

text: functions

text: after

text: adding

text: the

text: sparse

text: cutting-planes.

text: Our

text: ﬁrst

text: observation

text: is

text: a

text: negative

text: result:

text: super

text: sparse

text: cuts

text: as

text: deﬁned

text: for

text: the

text: packing-type

text: problems

text: can

text: be

text: arbitrarily

text: bad

text: for

text: the

text: case

text: of

text: covering

text: problems.

text: In

text: order

text: to

text: present

text: this

text: result,

text: let

text: formalize

text: the

text: notion

text: of

text: super

text: sparse

text: cuts

text: in

text: this

text: setting.

text: In

text: particular,

text: given

text: an

text: J

text: instance

text: of

text: type

text: (C),

text: we

text: assume

text: we

text: partition

text: the

text: variable

text: indices

text: n

text: into

text: J

text: J

text: J

text: q

text: For

text: all

text: j

text: [q]

text: we

text: add

text: all

text: possible

text: cuts

text: that

text: have

text: support

text: on

text: variables

text: with

text: index

text: in

text: J

text: Let

text: j

text: zS.S.

text: be

text: the

text: optimal

text: objective

text: function

text: of

text: the

text: resulting

text: LP

text: with

text: cuts.

text: Theorem

text: For

text: any

text: constant

text: M

text: there

text: exists

text: a

text: covering

text: integer

text: program

text: (C)

text: and

text: J

text: partition

text: J

text: J

text: of

text: [n],

text: such

text: that

text: the

text: corresponding

text: zS.S.

text: and

text: zI

text: satisﬁes:

text: zI

text: M

text: zS.S..

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: We now present the main result of this section. In particular, we present a worst-case upper bound on VzI that is independent of the data A, b, c, and depends only on GcovIer and the z ,C V A, V choice of the row support list . We remind the reader that given a graph G and collection V V of its vertices, η¯ (G) is the mixed chromatic number with respect to (see Deﬁnition 8). I ⊆ Theorem 22. Consider a covering integer programming as deﬁned in (C). Let 2[m] be a partition of the index set of rows of A and let G = GcovIer = (V, E) be the covering interaction V ⊆ A, graph of A. Then for any sparse cut support list 2V we have V ≥ 1 · z ,C V zI. η¯ (G) We make a few comments regarding Theorem 18: 1. While the result of Theorem 22 for covering-type IPs is very “similar” to the result of Theorem 11 for packing-type of IPs, the key ideas in the proofs are diﬀerent. 2. Like the previous discussion in Section 2.1, the chromatic numbers is small for graphs with small max degree. In fact, using Brook’s Theorem [5], we can obtain a result very similar to Corollary 12 for the covering case as well. 3. The result of Theorem 18 holds even if upper bounds are present on some or all of the variables (in this case, we also need to assume that the instance is feasible). I Consider the case of two-stage covering stochastic problem with K scenario and as deﬁned in Example 21. Since GcovIer is a clique, its chromatic number is K. Therefore we obtain the A, following corollary of Theorem 22. Corollary 23 (Strength of weak speciﬁc-scenario cuts for covering stochastic problems). Con- ∗ sider a two-stage covering stochastic problem for K scenario. Let z be the optimal objective value obtained after adding all weak speciﬁc-scenario cuts. Then ∗ ≥ 1 z zI. K Next we prove that the bound presented in Corollary 23 is tight (and therefore the result of Theorem 18 is tight for GcovIer being a clique). A, ∗ Theorem 24. Let z be the optimal objective value obtained after adding all weak speciﬁc- scenario cuts for a two-stage covering stochastic problem. Given any (cid:15) > 0 with (cid:15) < K, there exists an instance of the covering-type two-stage stochastic problem with K scenarios such that ∗ ≤ 1 · z − zI. (K (cid:15)) The proof of Theorem 24 is perhaps the most involved in this paper, as the family of instances constructed to prove the above theorem are signiﬁcantly complicated. All proofs of the above results are presented in Section 2.1. 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: now

text: present

text: the

text: main

text: result

text: of

text: this

text: section.

text: In

text: particular,

text: we

text: present

text: a

text: worst-case

text: upper

text: bound

text: on

text: VzI

text: that

text: is

text: independent

text: of

text: the

text: data

text: A,

text: b,

text: c,

text: and

text: depends

text: only

text: on

text: GcovIer

text: and

text: the

text: z

text: ,C

text: V

text: A,

text: V

text: choice

text: of

text: the

text: row

text: support

text: list

text: We

text: remind

text: the

text: reader

text: that

text: given

text: a

text: graph

text: G

text: and

text: collection

text: V

text: V

text: of

text: its

text: vertices,

text: η¯

text: (G)

text: is

text: the

text: mixed

text: chromatic

text: number

text: with

text: respect

text: to

text: (see

text: Deﬁnition

text: I

text: Theorem

text: Consider

text: a

text: covering

text: integer

text: programming

text: as

text: deﬁned

text: in

text: (C).

text: Let

text: be

text: a

text: partition

text: of

text: the

text: index

text: set

text: of

text: rows

text: of

text: A

text: and

text: let

text: G

text: GcovIer

text: (V,

text: E)

text: be

text: the

text: covering

text: interaction

text: V

text: A,

text: graph

text: of

text: A.

text: Then

text: for

text: any

text: sparse

text: cut

text: support

text: list

text: we

text: have

text: V

text: z

text: ,C

text: V

text: zI.

text: η¯

text: (G)

text: We

text: make

text: a

text: few

text: comments

text: regarding

text: Theorem

text: While

text: the

text: result

text: of

text: Theorem

text: for

text: covering-type

text: IPs

text: is

text: very

text: “similar”

text: to

text: the

text: result

text: of

text: Theorem

text: for

text: packing-type

text: of

text: IPs,

text: the

text: key

text: ideas

text: in

text: the

text: proofs

text: are

text: diﬀerent.

text: Like

text: the

text: previous

text: discussion

text: in

text: Section

text: the

text: chromatic

text: numbers

text: is

text: small

text: for

text: graphs

text: with

text: small

text: max

text: degree.

text: In

text: fact,

text: using

text: Brook’s

text: Theorem

text: we

text: can

text: obtain

text: a

text: result

text: very

text: similar

text: to

text: Corollary

text: for

text: the

text: covering

text: case

text: as

text: well.

text: The

text: result

text: of

text: Theorem

text: holds

text: even

text: if

text: upper

text: bounds

text: are

text: present

text: on

text: some

text: or

text: all

text: of

text: the

text: variables

text: (in

text: this

text: case,

text: we

text: also

text: need

text: to

text: assume

text: that

text: the

text: instance

text: is

text: feasible).

text: I

text: Consider

text: the

text: case

text: of

text: two-stage

text: covering

text: stochastic

text: problem

text: with

text: K

text: scenario

text: and

text: as

text: deﬁned

text: in

text: Example

text: Since

text: GcovIer

text: is

text: a

text: clique,

text: its

text: chromatic

text: number

text: is

text: K.

text: Therefore

text: we

text: obtain

text: the

text: A,

text: following

text: corollary

text: of

text: Theorem

text: Corollary

text: (Strength

text: of

text: weak

text: speciﬁc-scenario

text: cuts

text: for

text: covering

text: stochastic

text: problems).

text: Con-

text: sider

text: a

text: two-stage

text: covering

text: stochastic

text: problem

text: for

text: K

text: scenario.

text: Let

text: z

text: be

text: the

text: optimal

text: objective

text: value

text: obtained

text: after

text: adding

text: all

text: weak

text: speciﬁc-scenario

text: cuts.

text: Then

text: z

text: zI.

text: K

text: Next

text: we

text: prove

text: that

text: the

text: bound

text: presented

text: in

text: Corollary

text: is

text: tight

text: (and

text: therefore

text: the

text: result

text: of

text: Theorem

text: is

text: tight

text: for

text: GcovIer

text: being

text: a

text: clique).

text: A,

text: Theorem

text: Let

text: z

text: be

text: the

text: optimal

text: objective

text: value

text: obtained

text: after

text: adding

text: all

text: weak

text: speciﬁc-

text: scenario

text: cuts

text: for

text: a

text: two-stage

text: covering

text: stochastic

text: problem.

text: Given

text: any

text: with

text: K,

text: there

text: exists

text: an

text: instance

text: of

text: the

text: covering-type

text: two-stage

text: stochastic

text: problem

text: with

text: K

text: scenarios

text: such

text: that

text: z

text: zI.

text: (K

text: The

text: proof

text: of

text: Theorem

text: is

text: perhaps

text: the

text: most

text: involved

text: in

text: this

text: paper,

text: as

text: the

text: family

text: of

text: instances

text: constructed

text: to

text: prove

text: the

text: above

text: theorem

text: are

text: signiﬁcantly

text: complicated.

text: All

text: proofs

text: of

text: the

text: above

text: results

text: are

text: presented

text: in

text: Section

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:88) (cid:88) problem: (DSC(q)) min x + y v v ∈{ } ∈{ } v 0,1 q v 0,1 q ≥ s.t. Aqx + Aqy 1 ∈ { } x, y 0, 1 2q We ﬁrst argue that DSC(q) preserves the gap between IP and LP from SSC(q). Lemma 49. The IP optimal value of DSC(q) is at least q, while the LP relaxation optimal value is at most 2.(cid:80) ≤ Proof. (zLP 2): By Theorem 48, there exists a feasible solution x¯ of the LP relaxation of ≤ SSC(q) such that x¯ 2. Then (x¯, 0) is a feasible solution of the LP relaxation of DSC(q), v v giving the desired bound. ≥ (zIP q): Assume by contradiction that (x¯, y¯) is a feasible solution of DSC(q) with objective function less than q. Note that if x¯ = y¯ = 1, then we may set y¯ = 0 and still obtain a v v v feasible solution with a better objective function value; similarly, if x¯ = 0 and y¯ = 1 we may v v set x¯ = 1 and y¯ = 0 and obtain a feasible solution with same objective value. Therefore, we v v may assume that y¯ = 0. In this case, x¯ is a feasible solution of SSC with objective function less than q, contradicting the statement of Theorem 48. J { } We consider a partition on the columns of DSC(q) into two blocks: = J , J where J 1 1 1 corresponding to variables x and J corresponding to variables y. To complete the proof of the 2 theorem it is suﬃcient to prove that the super sparse closure optimal value zS.S. of DSC(q) is equal to optimal LP value zLP of DSC(q). Lemma 50. zS.S = zLP for DSC(q). ∩ Proof. Let P be the integer set for DSC(q). Since P S.S = P (J1) P (J2), Observation 31 gives ∈ ∈ ∈ | ∈ | that (x¯, y¯) P S.S iﬀ (x¯, y¯) P LP and x¯ P I and y¯ P I . But since P is of covering-type | J1 J2 ∈ ∈ and SSC(q) is feasible, we have that P I = [0, 1]2q, and thus (x¯, y¯) P S.S. iﬀ (x¯, y¯) P LP . Ji This concludes the proof. 5.2.2 Proof of Theorem 22 Consider a covering problem (C). As in the packing case, there is an identiﬁcation of sets (cid:83) (cid:83) of nodes of GcovIer with sets of indices of variables (the “indices in the union of their support”), I A, { } namely if = I , I , . . . , I is the given row index partition and the nodes of GcovIer are { } 1 2 q { } A, ⊆ v , v , . . . , v , then the set of vertices v ∈ corresponds to the indices ∈ ∈ supp(A ) 1 2 q i i I i I r Ii r [n]. We will make use of this correspondence, and in order to make statements precise we use the → function usupp : 2V (GcAo,vIer) 2[n] to denote this correspondence; with slight abuse of notation, { } { } for a singleton set v we use usupp(v) instead of usupp( v ). ⊆ Given a set of vertices S V (GcovIer), let x(S) be the optimal solution of the covering problem A, ∈ { | | ∈ | } projected to the variables relative to S, namely x(S) argmin (c )T y y P I . ∈ R usupp(S) usupp(S) Also, let zeroS(x(S)) n denote the solution appended by zeros in the original space, namely ∈ ∈ zeroS(x(S)) = x(S) if i usupp(S) and zeroS(x(S)) = 0 if i / usupp(S). i i i 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: problem:

text: (DSC(q))

text: min

text: x

text: y

text: v

text: v

text: v

text: q

text: v

text: q

text: s.t.

text: Aqx

text: Aqy

font: unknown

text: x,

text: y

text: We

text: ﬁrst

text: argue

text: that

text: DSC(q)

text: preserves

text: the

text: gap

text: between

text: IP

text: and

text: LP

text: from

text: SSC(q).

text: Lemma

text: The

text: IP

text: optimal

text: value

text: of

text: DSC(q)

text: is

text: at

text: least

text: q,

text: while

text: the

text: LP

text: relaxation

text: optimal

text: value

text: is

text: at

text: most

text: Proof.

text: (zLP

text: By

text: Theorem

text: there

text: exists

text: a

text: feasible

text: solution

text: x¯

text: of

text: the

text: LP

text: relaxation

text: of

text: SSC(q)

text: such

text: that

text: x¯

text: Then

text: (x¯,

text: is

text: a

text: feasible

text: solution

text: of

text: the

text: LP

text: relaxation

text: of

text: DSC(q),

text: v

text: v

text: giving

text: the

text: desired

text: bound.

text: (zIP

text: q):

text: Assume

text: by

text: contradiction

text: that

text: (x¯,

text: y¯)

text: is

text: a

text: feasible

text: solution

text: of

text: DSC(q)

text: with

text: objective

text: function

text: less

text: than

text: q.

text: Note

text: that

text: if

text: x¯

text: y¯

text: then

text: we

text: may

text: set

text: y¯

text: and

text: still

text: obtain

text: a

text: v

text: v

text: v

text: feasible

text: solution

text: with

text: a

text: better

text: objective

text: function

text: value;

text: similarly,

text: if

text: x¯

text: and

text: y¯

text: we

text: may

text: v

text: v

text: set

text: x¯

text: and

text: y¯

text: and

text: obtain

text: a

text: feasible

text: solution

text: with

text: same

text: objective

text: value.

text: Therefore,

text: we

text: v

text: v

text: may

text: assume

text: that

text: y¯

text: In

text: this

text: case,

text: x¯

text: is

text: a

text: feasible

text: solution

text: of

text: SSC

text: with

text: objective

text: function

text: less

text: than

text: q,

text: contradicting

text: the

text: statement

text: of

text: Theorem

text: J

text: We

text: consider

text: a

text: partition

text: on

text: the

text: columns

text: of

text: DSC(q)

text: into

text: two

text: blocks:

text: J

text: J

text: where

text: J

text: corresponding

text: to

text: variables

text: x

text: and

text: J

text: corresponding

text: to

text: variables

text: y.

text: To

text: complete

text: the

text: proof

text: of

text: the

text: theorem

text: it

text: is

text: suﬃcient

text: to

text: prove

text: that

text: the

text: super

text: sparse

text: closure

text: optimal

text: value

text: zS.S.

text: of

text: DSC(q)

text: is

text: equal

text: to

text: optimal

text: LP

text: value

text: zLP

text: of

text: DSC(q).

text: Lemma

text: zS.S

text: zLP

text: for

text: DSC(q).

text: Proof.

text: Let

text: P

text: be

text: the

text: integer

text: set

text: for

text: DSC(q).

text: Since

text: P

text: S.S

text: P

text: P

text: Observation

text: gives

text: that

text: (x¯,

text: y¯)

text: P

text: S.S

text: iﬀ

text: (x¯,

text: y¯)

text: P

text: LP

text: and

text: x¯

text: P

text: I

text: and

text: y¯

text: P

text: I

text: But

text: since

text: P

text: is

text: of

text: covering-type

text: and

text: SSC(q)

text: is

text: feasible,

text: we

text: have

text: that

text: P

text: I

text: and

text: thus

text: (x¯,

text: y¯)

text: P

text: S.S.

text: iﬀ

text: (x¯,

text: y¯)

text: P

text: LP

text: Ji

text: This

text: concludes

text: the

text: proof.

text: Proof

text: of

text: Theorem

text: Consider

text: a

text: covering

text: problem

text: (C).

text: As

text: in

text: the

text: packing

text: case,

text: there

text: is

text: an

text: identiﬁcation

text: of

text: sets

text: of

text: nodes

text: of

text: GcovIer

text: with

text: sets

text: of

text: indices

text: of

text: variables

text: (the

text: “indices

text: in

text: the

text: union

text: of

text: their

text: support”),

text: I

text: A,

text: namely

text: if

text: I

text: I

text: I

text: is

text: the

text: given

text: row

text: index

text: partition

text: and

text: the

text: nodes

text: of

text: GcovIer

text: are

text: q

text: A,

text: v

text: v

text: v

text: then

text: the

text: set

text: of

text: vertices

text: v

text: corresponds

text: to

text: the

text: indices

text: supp(A

text: q

text: i

text: i

text: I

text: i

text: I

text: r

text: Ii

text: r

text: [n].

text: We

text: will

text: make

text: use

text: of

text: this

text: correspondence,

text: and

text: in

text: order

text: to

text: make

text: statements

text: precise

text: we

text: use

text: the

text: function

text: usupp

text: (GcAo,vIer)

text: to

text: denote

text: this

text: correspondence;

text: with

text: slight

text: abuse

text: of

text: notation,

text: for

text: a

text: singleton

text: set

text: v

text: we

text: use

text: usupp(v)

text: instead

text: of

text: usupp(

text: v

text: Given

text: a

text: set

text: of

text: vertices

text: S

text: V

text: (GcovIer),

text: let

text: x(S)

text: be

text: the

text: optimal

text: solution

text: of

text: the

text: covering

text: problem

text: A,

text: projected

text: to

text: the

text: variables

text: relative

text: to

text: S,

text: namely

text: x(S)

text: argmin

text: (c

text: )T

text: y

text: y

text: P

text: I

text: R

text: usupp(S)

text: usupp(S)

text: Also,

text: let

text: zeroS(x(S))

text: n

text: denote

text: the

text: solution

text: appended

text: by

text: zeros

text: in

text: the

text: original

text: space,

text: namely

text: zeroS(x(S))

text: x(S)

text: if

text: i

text: usupp(S)

text: and

text: zeroS(x(S))

text: if

text: i

text: usupp(S).

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: | | | | → Proof. Since [nn] = [n]n , let g : [n]n [nn] be any bijection between the two sets. For ∈ { | ∈ } G { | ∈ } j [n], deﬁne the set Gi = g(cid:12)(u)(cid:12) u [n]n, u = j . Deﬁne i = Gi j [n] . It is easy to j (cid:12) (cid:12) i (cid:83) j check the following properties:(cid:12) (cid:12) ∈ − { | ∈ } 1. Given i, for any j [n], Gi = nn 1 and n Gi = g(u) u [n]n = [nn]. j (cid:84) j=1 j ∈ G ∈ G ∈ G 2. For a selection G1 1, G2 2, . . . , Gn n, consider u = (j , j , . . . , j ). Then j1 j2 ∈ jn 1 2 n according to the deﬁnition, g(u) n Gi , so the intersection of these sets is non-empty. i=1 ji This concludes the proof. (cid:83) ∈ Z G G Lemma 54. Let n . Consider a collection 1, . . . , n satisfying the properties of Lemma G ⊆ G++ G 53, and consider ¯i i for i = 1, . . . , n. If the sets in n ¯i cover the whole of [nn], then ∈ G G i=1 there is i [n] such that ¯i = i. (cid:83) ∈ G \ G ∈ G \ G Proof. By contradiction, suppose there is G1 1 ¯1, . . . , G(cid:83)n n ¯n. Then part 2 ∈ j1 jn G of Lemma 53, there exists an element u n Gi , and since the sets in i partition [nn] G i=1 ji G this means that u is not covered by sets in ¯i, for all i; then n ¯i does not cover [nn], a i=1 contradiction. ≥ { } Now pick a prime number n max K, 2 . We will construct an instance with variables G G x , . . . , x and y , . . . , y , and each row-block will have nn constraints. Let 1, . . . , n be a 1 n2 1 K F F collection satisfying the properties from Lemma 53, and let , . . . , be an aﬃne n-design F 1 n ∈ F ∈ F (we remind the readers – each partitions [n2] with n-subsets, and for A , B with (cid:54) | ∩ | ≤ i i j i = j, A B 1); this aﬃne design will be used to “permute” the indices of the x variables from one set of covering constraints to the next (see example above). We consider the explicit F { } enumerati∈on i = Fi1, .. . , Fin . (cid:12)  (cid:12) For k [K], we deﬁne the set (cid:12) (cid:88) (cid:88)  (cid:12) (cid:12)   (cid:12) n ∈ { } · ≥ Pk = (x, y) 0, 1 n2+K Akj xj + 1 yk 1 , ∈ i=1 j Fk i { } { } where the set of vectors Ak ∈ is equal to the set of vectors χ (cid:48)∈ ; we note that it is j j Fk Gi(cid:48) j [n] i j not important which Ak is assigned to which χ . j Gi(cid:48) j Then the covering integer p(cid:88)rogram we co(cid:88)nsider is the following: n2 K · min (cid:88) x(cid:88)j + nn yk j=1 k=1 n · ≥ ∀ ∈ s.t. Akj xj + 1 yk 1 k [K] ∈ i=1 j Fk i (cid:84) ∈ { } (x, y) 0, 1 n2+K. ∈ (or equivalently (x, y) ∈ P ). Let P denote the set of solutions for this problem. k [K] k 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Since

text: [nn]

text: [n]n

text: let

text: g

text: [n]n

text: [nn]

text: be

text: any

text: bijection

text: between

text: the

text: two

text: sets.

text: For

text: G

text: j

text: [n],

text: deﬁne

text: the

text: set

text: Gi

text: u

text: [n]n,

text: u

text: j

text: Deﬁne

text: i

text: Gi

text: j

text: [n]

text: It

text: is

text: easy

text: to

text: j

text: i

text: j

text: check

text: the

text: following

text: Given

text: i,

text: for

text: any

text: j

text: [n],

text: Gi

text: nn

text: and

text: n

text: Gi

text: g(u)

text: u

text: [n]n

text: [nn].

text: j

text: j

text: G

text: G

text: G

text: For

text: a

text: selection

text: Gn

text: n,

text: consider

text: u

text: (j

text: j

text: j

text: Then

text: jn

text: n

text: according

text: to

text: the

text: deﬁnition,

text: g(u)

text: n

text: Gi

text: so

text: the

text: intersection

text: of

text: these

text: sets

text: is

text: non-empty.

text: ji

text: This

text: concludes

text: the

text: proof.

text: Z

text: G

text: G

text: Lemma

text: Let

text: n

text: Consider

text: a

text: collection

text: n

text: satisfying

text: the

text: properties

text: of

text: Lemma

text: G

text: G++

text: G

text: and

text: consider

text: ¯i

text: i

text: for

text: i

text: n.

text: If

text: the

text: sets

text: in

text: n

text: ¯i

text: cover

text: the

text: whole

text: of

text: [nn],

text: then

text: G

text: G

text: there

text: is

text: i

text: [n]

text: such

text: that

text: ¯i

text: i.

text: G

text: G

text: G

text: G

text: Proof.

text: By

text: contradiction,

text: suppose

text: there

text: is

text: n

text: ¯n.

text: Then

text: part

text: jn

text: G

text: of

text: Lemma

text: there

text: exists

text: an

text: element

text: u

text: n

text: Gi

text: and

text: since

text: the

text: sets

text: in

text: i

text: partition

text: [nn]

text: G

text: ji

text: G

text: this

text: means

text: that

text: u

text: is

text: not

text: covered

text: by

text: sets

text: in

text: ¯i,

text: for

text: all

text: i;

text: then

text: n

text: ¯i

text: does

text: not

text: cover

text: [nn],

text: a

text: contradiction.

text: Now

text: pick

text: a

text: prime

text: number

text: n

text: max

text: K,

text: We

text: will

text: construct

text: an

text: instance

text: with

text: variables

text: G

text: G

text: x

text: x

text: and

text: y

text: y

text: and

text: each

text: row-block

text: will

text: have

text: nn

text: constraints.

text: Let

text: n

text: be

text: a

text: K

text: F

text: F

text: collection

text: satisfying

text: the

text: properties

text: from

text: Lemma

text: and

text: let

text: be

text: an

text: aﬃne

text: n-design

text: F

text: n

text: F

text: F

text: (we

text: remind

text: the

text: readers

text: each

text: partitions

text: with

text: n-subsets,

text: and

text: for

text: A

text: B

text: with

text: i

text: i

text: j

text: i

text: j,

text: A

text: B

text: this

text: aﬃne

text: design

text: will

text: be

text: used

text: to

text: “permute”

text: the

text: indices

text: of

text: the

text: x

text: variables

text: from

text: one

text: set

text: of

text: covering

text: constraints

text: to

text: the

text: next

text: (see

text: example

text: above).

text: We

text: consider

text: the

text: explicit

text: F

text: enumerati∈on

text: i

text: Fin

text: For

text: k

text: [K],

text: we

text: deﬁne

text: the

text: set

text: n

text: Pk

text: (x,

text: y)

text: Akj

text: xj

text: yk

text: j

text: Fk

text: i

text: where

text: the

text: set

text: of

text: vectors

text: Ak

text: is

text: equal

text: to

text: the

text: set

text: of

text: vectors

text: χ

text: we

text: note

text: that

text: it

text: is

text: j

text: j

text: Fk

text: j

text: [n]

text: i

text: j

text: not

text: important

text: which

text: Ak

text: is

text: assigned

text: to

text: which

text: χ

text: j

text: j

text: Then

text: the

text: covering

text: integer

text: we

text: is

text: the

text: following:

text: K

text: min

text: nn

text: yk

text: n

text: s.t.

text: Akj

text: xj

text: yk

text: k

text: [K]

text: j

text: Fk

text: i

text: (x,

text: y)

text: (or

text: equivalently

text: (x,

text: y)

text: P

text: Let

text: P

text: denote

text: the

text: set

text: of

text: solutions

text: for

text: this

text: problem.

text: k

text: [K]

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:80) ∈ ∈ It is straightforward to verify that (xu, y) P for u [n]. Thus, the average 1 ∈ (xu, y) | | n u [n] belongs to P I. It then follows that (x¯, y¯k) = (n1 1, 0) (x,yk) belongs to P I (x,yk). This concludes the proof. ≥ − − Putting Lemmas 56 and 57 together we get zI Kn K2 = K K2 . For large enough n, ≥ − zS.S. n n we get VzI K (cid:15). This concludes the proof of Theorem 24. z ,C 5.3 Proof for packing-type problem with arbitrary A matrix 5.3.1 Proof of Theorem 26 J In this section, we use the same notation as that used in Section 2.1. So, let be a { } partition of the index set of columns of A (that is [n]). Let V = v , . . . , v be the vertices of 1 q (cid:83) (cid:80) V { } ⊆ V GpaJck (based on Deﬁnition 4). Let ˜ = V u1, V u2, . . . , V uk be the subset of sparse cut A, support list corresponding to the deﬁnition of corrected average density DV (see Deﬁnition 25), | | (cid:83) so V = k V ui and 1 k V ui = DV. i=1 k i=1 (cid:8) (cid:9) Recall from Section 2.1 a couple of deﬁnitions: ﬁrst, the function φ maps subsets of vertices { } of V to the corresponding variable indices, namely if S = v ∈ then φ(S) = ∈ J . | i i I | | ∈ i I i (cid:80) For the purpose of this section, let x(S) = argmax (c )T (x ) x P I . (Notice that φ(S) φ(S) this is diﬀerent from the deﬁnition used in Section 2.1.) ∈ V V ≤ Lemma 58. For any set V˜ ˜ we have z ,P cT x(V˜) + ∈ \ cT x(v). v(cid:84)V V˜ ∈ V ⊆ ∗ { | ∈ V } Proof. Fix any V˜ ˜ and S V˜ . Let x = argmax cT x x P ,P be an optimal solution V V (cid:48) ∗ ∈ corresponding to the optimization over P ,P . Since P ,C = (cid:48)∈V P (V˜ ), we have that x ⊇ (cid:48) ∗| ∈ | V˜ ˜ P (S) P (V˜ ). From Observation 31 we then get x P I . | ∗| ≤ | |φ(S) ≤ φ(S) Thus we get (c )T (x ) (c )T (x(S) ) cT x(S), where the ﬁrst inequality φ(S) φ(S) φ(S) φ(S) follows from optimality of x(S) and the second inequality follows from non-negativity of c and x(S). V { } In particular, since ˜ covers V , we can apply this to the any singleton S = v and get | ∗| ≤ (c )T (x ) cT x(v). φ(v) φ(v) (cid:88) ∈ V (cid:88) Applying this bound, we obtain that for any V˜ V ∗ | ∗| | ∗| ≤ z ,P = cT x = (c )T (x ) + (c )T (x ) cT x(V˜) + cT x(v). φ(V˜) φ(V˜) φ(v) φ(v) ∈ \ ∈ \ v V V˜ v V V˜ This concludes the proof. V Now we are ready to complete the proof of the theorem. Using Lemma 58 for all sets in ˜ and adding up these inequalities we obtain that   (cid:88) (cid:88) (cid:88)   k k · V ≤ k z ,P cT x(V ui) + cT x(v) (cid:88) (cid:88) ∈ \ i=1 i=1 v V V ui k · = cT x(V ui) + miss(v) cT x(v), (11) ∈ i=1 v V 36 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: It

text: is

text: straightforward

text: to

text: verify

text: that

text: (xu,

text: y)

text: P

text: for

text: u

text: [n].

text: Thus,

text: the

text: average

text: (xu,

text: y)

text: n

text: u

text: [n]

text: belongs

text: to

text: P

text: I.

text: It

text: then

text: follows

text: that

text: (x¯,

text: y¯k)

text: (x,yk)

text: belongs

text: to

text: P

text: I

text: (x,yk).

text: This

text: concludes

text: the

text: proof.

text: Putting

text: Lemmas

text: and

text: together

text: we

text: get

text: zI

text: Kn

text: K

text: For

text: large

text: enough

text: n,

text: zS.S.

text: n

text: n

text: we

text: get

text: VzI

text: K

text: This

text: concludes

text: the

text: proof

text: of

text: Theorem

text: z

text: ,C

label: section

text: Proof

text: for

text: packing-type

text: problem

text: with

text: arbitrary

text: A

text: matrix

text: Proof

text: of

text: Theorem

text: J

text: In

text: this

text: section,

text: we

text: use

text: the

text: same

text: notation

text: as

text: that

text: used

text: in

text: Section

text: So,

text: let

text: be

text: a

text: partition

text: of

text: the

text: index

text: set

text: of

text: columns

text: of

text: A

text: (that

text: is

text: [n]).

text: Let

text: V

text: v

text: v

text: be

text: the

text: vertices

text: of

text: q

text: V

text: V

text: GpaJck

text: (based

text: on

text: Deﬁnition

text: Let

text: V

text: V

text: V

text: uk

text: be

text: the

text: subset

text: of

text: sparse

text: cut

text: A,

text: support

text: list

text: corresponding

text: to

text: the

text: deﬁnition

text: of

text: corrected

text: average

text: density

text: DV

text: (see

text: Deﬁnition

text: so

text: V

text: k

text: V

text: ui

text: and

text: k

text: V

text: ui

text: DV.

text: k

text: Recall

text: from

text: Section

text: a

text: couple

text: of

text: deﬁnitions:

text: ﬁrst,

text: the

text: function

text: φ

text: maps

text: subsets

text: of

text: vertices

text: of

text: V

text: to

text: the

text: corresponding

text: variable

text: indices,

text: namely

text: if

text: S

text: v

text: then

text: φ(S)

text: J

text: i

text: i

text: I

text: i

text: I

text: i

text: For

text: the

text: purpose

text: of

text: this

text: section,

text: let

text: x(S)

text: argmax

text: (c

text: )T

text: (x

text: x

text: P

text: I

text: (Notice

text: that

text: φ(S)

text: φ(S)

text: this

text: is

text: diﬀerent

text: from

text: the

text: deﬁnition

text: used

text: in

text: Section

text: V

text: V

text: Lemma

text: For

text: any

text: set

text: V˜

text: we

text: have

text: z

text: ,P

text: cT

text: x(V˜)

text: cT

text: x(v).

text: V˜

text: V

text: V

text: Proof.

text: Fix

text: any

text: V˜

text: and

text: S

text: V˜

text: Let

text: x

text: argmax

text: cT

text: x

text: x

text: P

text: ,P

text: be

text: an

text: optimal

text: solution

text: V

text: V

text: corresponding

text: to

text: the

text: optimization

text: over

text: P

text: ,P

text: Since

text: P

text: ,C

text: P

text: (V˜

text: we

text: have

text: that

text: x

text: V˜

text: P

text: (S)

text: P

text: (V˜

text: From

text: Observation

text: we

text: then

text: get

text: x

text: P

text: I

text: |φ(S)

text: φ(S)

text: Thus

text: we

text: get

text: (c

text: )T

text: (x

text: (c

text: )T

text: (x(S)

text: cT

text: x(S),

text: where

text: the

text: ﬁrst

text: inequality

text: φ(S)

text: φ(S)

text: φ(S)

text: φ(S)

text: follows

text: from

text: optimality

text: of

text: x(S)

text: and

text: the

text: second

text: inequality

text: follows

text: from

text: non-negativity

text: of

text: c

text: and

text: x(S).

text: V

text: In

text: particular,

text: since

text: covers

text: V

text: we

text: can

text: apply

text: this

text: to

text: the

text: any

text: singleton

text: S

text: v

text: and

text: get

text: (c

text: )T

text: (x

text: cT

text: x(v).

text: φ(v)

text: φ(v)

text: V

text: Applying

text: this

text: bound,

text: we

text: obtain

text: that

text: for

text: any

text: V˜

text: V

text: z

text: ,P

text: cT

text: x

text: (c

text: )T

text: (x

text: (c

text: )T

text: (x

text: cT

text: x(V˜)

text: cT

text: x(v).

text: φ(V˜)

text: φ(V˜)

text: φ(v)

text: φ(v)

text: v

text: V

text: V˜

text: v

text: V

text: V˜

text: This

text: concludes

text: the

text: proof.

text: V

text: Now

text: we

text: are

text: ready

text: to

text: complete

text: the

text: proof

text: of

text: the

text: theorem.

text: Using

text: Lemma

text: for

text: all

text: sets

text: in

text: and

text: adding

text: up

text: these

text: inequalities

text: we

text: obtain

text: that

text: k

text: k

text: V

text: k

text: z

text: ,P

text: cT

text: x(V

text: ui)

text: cT

text: x(v)

text: v

text: V

text: V

text: ui

text: k

text: cT

text: x(V

text: ui)

text: miss(v)

text: cT

text: x(v),

text: v

text: V

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Let P denote the associated integer set. The partition of variable indices we consider is J { } = J , J , J , . . . , J , where J corresponds to the variables x, and each J corresponds 0 1 3 ∈ K 0 k to variable y for k [K]. Notice that the packing interaction graph GpaJck is a star on K + 1 k A, { } {{ } { }} nodes; explicitly, GpaJck = (V, E) with V = v , . . . , v and E = v , v , . . . , v , v . Re- A, 0 K V0 1 0 K call we are in the natural sparse closure setting, so the support list in this case equals the edge set E. ≥ · We show that zN.S. K zI. For that, we start by lower bounding zN.S.. ≥ Lemma 61. zN.S. K. (cid:84) Proof. We show that the solution (x¯, y¯) given by x¯ = 11 and y¯ = 1 belongs to P N.S.. Following ∈ 2 ∈ ∈ | Observation 31, to show (x¯, y¯) P N.S. it suﬃces to show (x¯, y¯) P LP and (x¯, y¯ ) P I ∈ | | k (x,yk) for all k [K]. Notice that P LP = ∈ P I and P I = P I (the latter uses the fact | (cid:54) k [K] k (x,yk) ∈k (x,yk) ∈ P I = [0, 1]K+1 for j = k). Thus it suﬃces to show (x¯, y¯) P I for all k [K] j (x,yk) ∈ k For that, ﬁx k [K] and consider th(cid:26)e points (xk1, ek) and (xk2, ek), where ei is the ith canonical basis vector and (cid:26) 1 if i = k, xk1 = i 0 otherwise 0 if i = k, xk2 = . i 1 otherwise By deﬁnition both these points belong to Pk; the average 12(xk1, ek) + 12(xk2, ek) = (121, ek) also belongs to P . Moreover, since the constraints deﬁning P I are independent of variable y for (cid:54) k k i i = k, we have that (11, 1) = (x¯, y¯) also belongs to P I. This concludes the proof. 2 k Now it is easy to see from the deﬁnition of P that no feasible solution to the IP can set k more than one y variable to 1, and hence the optimal value zi is at most 1. Together with the ≥ · previous lemma, this gives the desired bound zN.S. K zI, thus concluding the proof of the theorem. 6 Acknowledgements We would like to thank Carla Michini for her comments that helped improve the presentation in this paper. Santanu S. Dey and Qianyi Wang acknowledge the support from NSF CMMI Grant 1149400. References [1] Acer, S., Kayaaslan, E., Aykanat, C.: A recursive bipartitioning algorithm for permuting sparse square matrices into block diagonal form with overlap. SIAM J. Scientiﬁc Computing 35(1) (2013) [2] Bergner, M., Caprara, A., Furini, F., Lu¨bbecke, M.E., Malaguti, E., Traversi, E.: Partial convexiﬁcation of general mips by dantzig-wolfe reformulation. In: O. Gu¨nlu¨k, G.J. Woeg- inger (eds.) Integer Programming and Combinatoral Optimization - 15th International Conference, IPCO 2011, New York, NY, USA, June 15-17, 2011. Proceedings, Lecture Notes in Computer Science, vol. 6655, pp. 39–51. Springer (2011) 39 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

text: P

text: denote

text: the

text: associated

text: integer

text: set.

text: The

text: partition

text: of

text: variable

text: indices

text: we

text: consider

text: is

text: J

text: J

text: J

text: J

text: J

text: where

text: J

text: corresponds

text: to

text: the

text: variables

text: x,

text: and

text: each

text: J

text: corresponds

text: K

text: k

text: to

text: variable

text: y

text: for

text: k

text: [K].

text: Notice

text: that

text: the

text: packing

text: interaction

text: graph

text: GpaJck

text: is

text: a

text: star

text: on

text: K

text: k

text: A,

text: nodes;

text: explicitly,

text: GpaJck

text: (V,

text: E)

text: with

text: V

text: v

text: v

text: and

text: E

text: v

text: v

text: v

text: v

text: Re-

text: A,

text: K

text: K

text: call

text: we

text: are

text: in

text: the

text: natural

text: sparse

text: closure

text: setting,

text: so

text: the

text: support

text: list

text: in

text: this

text: case

text: equals

text: the

text: edge

text: set

text: E.

text: We

text: show

text: that

text: zN.S.

text: K

text: zI.

text: For

text: that,

text: we

text: start

text: by

text: lower

text: bounding

text: zN.S..

text: Lemma

text: zN.S.

text: K.

text: Proof.

text: We

text: show

text: that

text: the

text: solution

text: (x¯,

text: y¯)

text: given

text: by

text: x¯

text: and

text: y¯

text: belongs

text: to

text: P

text: N.S..

text: Following

text: Observation

text: to

text: show

text: (x¯,

text: y¯)

text: P

text: N.S.

text: it

text: suﬃces

text: to

text: show

text: (x¯,

text: y¯)

text: P

text: LP

text: and

text: (x¯,

text: y¯

text: P

text: I

text: k

text: (x,yk)

text: for

text: all

text: k

text: [K].

text: Notice

text: that

text: P

text: LP

text: P

text: I

text: and

text: P

text: I

text: P

text: I

text: (the

text: latter

text: uses

text: the

text: fact

text: k

text: [K]

text: k

text: (x,yk)

text: ∈k

text: (x,yk)

text: P

text: I

text: for

text: j

text: k).

text: Thus

text: it

text: suﬃces

text: to

text: show

text: (x¯,

text: y¯)

text: P

text: I

text: for

text: all

text: k

text: [K]

text: j

text: (x,yk)

text: k

text: For

text: that,

text: ﬁx

text: k

text: [K]

text: and

text: consider

text: points

text: ek)

text: and

text: ek),

text: where

text: ei

text: is

text: the

text: ith

text: canonical

text: basis

text: vector

text: and

text: if

text: i

text: k,

text: i

text: otherwise

text: if

text: i

text: k,

text: i

text: otherwise

text: By

text: deﬁnition

text: both

text: these

text: points

text: belong

text: to

text: Pk;

text: the

text: average

text: ek)

text: ek)

text: ek)

text: also

text: belongs

text: to

text: P

text: Moreover,

text: since

text: the

text: constraints

text: deﬁning

text: P

text: I

text: are

text: independent

text: of

text: variable

text: y

text: for

text: k

text: k

text: i

text: i

text: k,

text: we

text: have

text: that

text: (x¯,

text: y¯)

text: also

text: belongs

text: to

text: P

text: I.

text: This

text: concludes

text: the

text: proof.

text: k

text: Now

text: it

text: is

text: easy

text: to

text: see

text: from

text: the

text: deﬁnition

text: of

text: P

text: that

text: no

text: feasible

text: solution

text: to

text: the

text: IP

text: can

text: set

text: k

text: more

text: than

text: one

text: y

text: variable

text: to

text: and

text: hence

text: the

text: optimal

text: value

text: zi

text: is

text: at

text: most

text: Together

text: with

text: the

text: previous

text: lemma,

text: this

text: gives

text: the

text: desired

text: bound

text: zN.S.

text: K

text: zI,

text: thus

text: concluding

text: the

text: proof

text: of

text: the

text: theorem.

text: Acknowledgements

text: We

text: would

text: like

text: to

text: thank

text: Carla

text: Michini

text: for

text: her

text: comments

text: that

text: helped

text: improve

text: the

text: presentation

text: in

text: this

text: paper.

text: Santanu

text: S.

text: Dey

text: and

text: Qianyi

text: Wang

text: acknowledge

text: the

text: support

text: from

text: NSF

text: CMMI

text: Grant

text: References

text: Acer,

text: S.,

text: Kayaaslan,

text: E.,

text: Aykanat,

text: C.:

text: A

text: recursive

text: bipartitioning

text: algorithm

text: for

text: permuting

text: sparse

text: square

text: matrices

text: into

text: block

text: diagonal

text: form

text: with

text: overlap.

text: SIAM

text: J.

text: Scientiﬁc

text: Computing

text: Bergner,

text: M.,

text: Caprara,

text: A.,

text: Furini,

text: F.,

text: Lu¨bbecke,

text: M.E.,

text: Malaguti,

text: E.,

text: Traversi,

text: E.:

text: Partial

text: convexiﬁcation

text: of

text: general

text: mips

text: by

text: dantzig-wolfe

text: reformulation.

text: In:

text: O.

text: Gu¨nlu¨k,

text: G.J.

text: Woeg-

text: inger

text: (eds.)

text: Integer

text: Programming

text: and

text: Combinatoral

text: Optimization

text: International

text: Conference,

text: IPCO

text: New

text: York,

text: NY,

text: USA,

text: June

text: Proceedings,

text: Lecture

text: Notes

text: in

text: Computer

text: Science,

text: vol.

text: pp.

text: Springer

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: [20] Molloy, M., Reed, B.: Graph Colouring and the Probabilistic Method. Springer-Verlag Berlin Heidelberg (2002) [21] Richard, J.P.P., Dey, S.S.: The group-theoretic approach in mixed integer programming. In: Ju¨nger et al. [14], chap. 19, pp. 727–801 [22] Walter, M.: Sparsity of lift-and-project cutting planes. In: S. Helber, M. Breitner, D. Rsch, C. Schn, J.M. Graf von der Schulenburg, P. Sibbertsen, M. Steinbach, S. Weber, A. Wolter (eds.) Operations Research Proceedings 2012, Operations Research Proceedings, pp. 9–14. Springer International Publishing (2012) [23] Wang, J., Ralphs, T.K.: Computational experience with hypergraph-based methods for automatic decomposition in discrete optimization. In: C.P. Gomes, M. Sellmann (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Opti- mization Problems, 10th International Conference, CPAIOR 2013, Yorktown Heights, NY, USA, May 18-22, 2013. Proceedings, Lecture Notes in Computer Science, vol. 7874, pp. 394–402. Springer (2013) [24] Zhang, M., Ku¨¸cu¨kyavuz, S.: Finitely convergent decomposition algorithms for two-stage stochastic pure integer programs. SIAM Journal on Optimization 24(4), 1933–1951 A Upper bound on zcut Assume we have the general formulation max cT x ≤ s.t. Ax b ∈ B x n, ∈ R × where A m n. Recall that we are interested in three type of problems: packing, covering (cid:84) and packing with arbitrary constraint matrix. All these categories will be written in the form of { ∈ | (cid:54) } the formulation above with diﬀerent restrictions on A and c. Let N = j [n] A = 0 be the N { i } N ij index set of non-zero entries of ith row of A. Let = N1, N2, . . . , Nt , denote P = ti=1 P (Ni) and zcut = max ∈ N cT x. x P Our basic strategy is the following: we keep adding cuts on the support of some N and i checking whether the LP solution will improve. We stop adding cuts when the objective function value does not change, thus obtaining an upper bound on zcut. The formal algorithm is shown as Algorithm 1. Once we stop adding cut on some N , we check whether there is a valid cut on N . The i i+1 index count is the number of groups of supports that adding cuts will not improve the optimal objective function value. Also, as long as adding a cut produces improvement on the objective value, count will be reset as 0. The algorithm terminates when one of the following happens: 1. An integral feasible solution is found. 2. The parameter count equals to the number of supports t. 41

label: reference

text: Molloy,

text: M.,

text: Reed,

text: B.:

text: Graph

text: Colouring

text: and

text: the

text: Probabilistic

text: Method.

text: Springer-Verlag

text: Berlin

text: Heidelberg

text: Richard,

text: J.P.P.,

text: Dey,

text: S.S.:

text: The

text: group-theoretic

text: approach

text: in

text: mixed

text: integer

text: programming.

text: In:

text: Ju¨nger

text: et

text: al.

text: chap.

text: pp.

text: Walter,

text: M.:

text: Sparsity

text: of

text: lift-and-project

text: cutting

text: planes.

text: In:

text: S.

text: Helber,

text: M.

text: Breitner,

text: D.

text: Rsch,

text: C.

text: Schn,

text: J.M.

text: Graf

text: von

text: der

text: Schulenburg,

text: P.

text: Sibbertsen,

text: M.

text: Steinbach,

text: S.

text: Weber,

text: A.

text: Wolter

text: (eds.)

text: Operations

text: Research

text: Proceedings

text: Operations

text: Research

text: Proceedings,

text: pp.

text: Springer

text: International

text: Publishing

text: Wang,

text: J.,

text: Ralphs,

text: T.K.:

text: Computational

text: experience

text: with

text: hypergraph-based

text: methods

text: for

text: automatic

text: decomposition

text: in

text: discrete

text: optimization.

text: In:

text: C.P.

text: Gomes,

text: M.

text: Sellmann

text: (eds.)

text: Integration

text: of

text: AI

text: and

text: OR

text: Techniques

text: in

text: Constraint

text: Programming

text: for

text: Combinatorial

text: Opti-

text: mization

text: Problems,

text: International

text: Conference,

text: CPAIOR

text: Yorktown

text: Heights,

text: NY,

text: USA,

text: May

text: Proceedings,

text: Lecture

text: Notes

text: in

text: Computer

text: Science,

text: vol.

text: pp.

text: Springer

text: Zhang,

text: M.,

text: Ku¨¸cu¨kyavuz,

text: S.:

text: Finitely

text: convergent

text: decomposition

text: algorithms

text: for

text: two-stage

text: stochastic

text: pure

text: integer

text: programs.

text: SIAM

text: Journal

text: on

text: Optimization

text: A

text: Upper

text: bound

text: on

text: zcut

text: Assume

text: we

text: have

text: the

text: general

text: formulation

text: max

text: cT

text: x

text: s.t.

text: Ax

text: b

text: B

text: x

text: n,

text: R

text: where

text: A

text: m

text: n.

text: Recall

text: that

text: we

text: are

text: interested

text: in

text: three

text: type

text: of

text: problems:

text: packing,

text: covering

text: and

text: packing

text: with

text: arbitrary

text: constraint

text: matrix.

text: All

text: these

text: categories

text: will

text: be

text: written

text: in

text: the

text: form

text: of

text: the

text: formulation

text: above

text: with

text: diﬀerent

text: restrictions

text: on

text: A

text: and

text: c.

text: Let

text: N

text: j

text: [n]

text: A

text: be

text: the

text: N

text: i

text: N

text: ij

text: index

text: set

text: of

text: non-zero

text: entries

text: of

text: ith

text: row

text: of

text: A.

text: Let

text: Nt

text: denote

text: P

text: P

text: (Ni)

text: and

text: zcut

text: max

text: N

text: cT

text: x.

text: x

text: P

text: Our

text: basic

text: strategy

text: is

text: the

text: following:

text: we

text: keep

text: adding

text: cuts

text: on

text: the

text: support

text: of

text: some

text: N

text: and

text: i

text: checking

text: whether

text: the

text: LP

text: solution

text: will

text: improve.

text: We

text: stop

text: adding

text: cuts

text: when

text: the

text: objective

text: function

text: value

text: does

text: not

text: change,

text: thus

text: obtaining

text: an

text: upper

text: bound

text: on

text: zcut.

text: The

text: formal

text: algorithm

text: is

text: shown

text: as

text: Algorithm

text: Once

text: we

text: stop

text: adding

text: cut

text: on

text: some

text: N

text: we

text: check

text: whether

text: there

text: is

text: a

text: valid

text: cut

text: on

text: N

text: The

text: i

text: index

text: count

text: is

text: the

text: number

text: of

text: groups

text: of

text: supports

text: that

text: adding

text: cuts

text: will

text: not

text: improve

text: the

text: optimal

text: objective

text: function

text: value.

text: Also,

text: as

text: long

text: as

text: adding

text: a

text: cut

text: produces

text: improvement

text: on

text: the

text: objective

text: value,

text: count

text: will

text: be

text: reset

text: as

text: The

text: algorithm

text: terminates

text: when

text: one

text: of

text: the

text: following

text: happens:

text: An

text: integral

text: feasible

text: solution

text: is

text: found.

text: The

text: parameter

text: count

text: equals

text: to

text: the

text: number

text: of

text: supports

text: t.


text: J {{ } { } { }} − − Consider the partition = 1 , 2 , 3 so that the graph GpacJk equals the path v v v . V {{ } { }} A, 2 1 3 Consider the support list = 1, 2 , 1, 3 for the “natural sparse closure” setting. Then V M {{ }} M {{ }} the maximal mixed stable sets of subordinate to are = 1, 2 , = 1, 3 and M {{ } { }} M {{ }} 1 2 = 2 , 3 ; = 1 is a non-maximal mixed stable set. 3 4 To see that mixed stable sets identify sets of variables that can be set independently and still yield a feasible solution, for i = 1, 2, 3 let x(i) be the optimal solution to the above packing (cid:54) (cid:98) (cid:99) problem conditioned on x = 0 for all j = i; for example x(2) = (0, b /a , 0) and x(3) = (cid:98) (cid:99) j M {{ } { }} 1 12 (0, 0, b /a ). Taking the mixed stable set = 2 , 3 we see that the combination of 2 23 (cid:98) (cid:99) (cid:98) (cid:99) 3 x(2) + x(3) = (0, b /a , b /a ) is also feasible for the problem. 1 12 2 23 V Moreover, these solutions allow us to upper bound the ratio z ,P /zI, namely the quality { of the column block-sparse closure. First, the integer optimum zI is at least max cT (x(2) + } V ≤ x(3)), cT x(1) . Also, one can show that z ,P cT (x(2) + x(3)) + cT x(1) (this uses the fact that actually x(2) + x(3) is the optimal solution for the problem conditioned on x = 0, and x(1) the V 1 ≤ optimal solution conditioned on x = x = 0). Together this gives z ,P /zI 2. Notice that V 2 3 M the upper bound on z is obtained by adding up the solutions corresponding to the sets and M 3 , which together cover all the variables of the problem. Looking at the fractional chromatic 4 V number η (GpacJk) allow us to provide essentially the best such bound. A, V Our ﬁrst result gives a worst-case upper bound on z ,P that is, surprisingly, independent zI of the data A, b, c, and depends only on the packing interaction graph GpaJck and the choice of V A, sparse cut support list . J ⊆ Theorem 11. Consider a packing integer program as deﬁned in (P). Let 2[n] be a partition of the index set of columns of A and let G = GpacJk = (V, E) be the packing interaction graph of V ⊆ A, A. Then for any sparse cut support list 2V we have V ≤ V · z ,P η (G) zI. V As discussed before, if we are considering the super sparse closure P S.S., η (G) is the usual fractional chromatic number. Therefore we obtain the following possibly weaker bound using Brook’s theorem [5]. J ⊆ Corollary 12. Consider a packing integer program as deﬁned in (P). Let 2[n] be a partition of the index set of columns of A and let GpacJk be the packing interaction graph of A. A, Let ∆ denote the maximum degree of G. Then we have the following bounds on the optimum value of the super sparse closure P S.S: 1. If GpacJk is not a complete graph or an odd cycle, then A, ≤ · zS.S. ∆ zI. 9 ##LTLine##

text: J

text: Consider

text: the

text: partition

text: so

text: that

text: the

text: graph

text: GpacJk

text: equals

text: the

text: path

text: v

text: v

text: v

text: V

text: A,

text: Consider

text: the

text: support

text: list

text: for

text: the

text: “natural

text: sparse

text: closure”

text: setting.

text: Then

text: V

text: M

text: M

text: the

text: maximal

text: mixed

text: stable

text: sets

text: of

text: subordinate

text: to

text: are

text: and

text: M

text: M

text: is

text: a

text: non-maximal

text: mixed

text: stable

text: set.

text: To

text: see

text: that

text: mixed

text: stable

text: sets

text: identify

text: sets

text: of

text: variables

text: that

text: can

text: be

text: set

text: independently

text: and

text: still

text: yield

text: a

text: feasible

text: solution,

text: for

text: i

text: let

text: x(i)

text: be

text: the

text: optimal

text: solution

text: to

text: the

text: above

text: packing

text: problem

text: conditioned

text: on

text: x

text: for

text: all

text: j

text: i;

text: for

text: example

text: b

text: /a

text: and

text: j

text: M

text: b

text: /a

text: Taking

text: the

text: mixed

text: stable

text: set

text: we

text: see

text: that

text: the

text: combination

text: of

text: b

text: /a

text: b

text: /a

text: is

text: also

text: feasible

text: for

text: the

text: problem.

text: V

text: Moreover,

text: these

text: solutions

text: allow

text: us

text: to

text: upper

text: bound

text: the

text: ratio

text: z

text: ,P

text: /zI,

text: namely

text: the

text: quality

text: of

text: the

text: column

text: block-sparse

text: closure.

text: First,

text: the

text: integer

text: optimum

text: zI

text: is

text: at

text: least

text: max

text: cT

text: V

text: cT

text: Also,

text: one

text: can

text: show

text: that

text: z

text: ,P

text: cT

text: cT

text: (this

text: uses

text: the

text: fact

text: that

text: actually

text: is

text: the

text: optimal

text: solution

text: for

text: the

text: problem

text: conditioned

text: on

text: x

text: and

text: the

text: V

text: optimal

text: solution

text: conditioned

text: on

text: x

text: x

text: Together

text: this

text: gives

text: z

text: ,P

text: /zI

text: Notice

text: that

text: V

text: M

text: the

text: upper

text: bound

text: on

text: z

text: is

text: obtained

text: by

text: adding

text: up

text: the

text: solutions

text: corresponding

text: to

text: the

text: sets

text: and

text: M

text: which

text: together

text: cover

text: all

text: the

text: variables

text: of

text: the

text: problem.

text: Looking

text: at

text: the

text: fractional

text: chromatic

text: V

text: number

text: η

text: (GpacJk)

text: allow

text: us

text: to

text: provide

text: essentially

text: the

text: best

text: such

text: bound.

text: A,

text: V

text: Our

text: ﬁrst

text: result

text: gives

text: a

text: worst-case

text: upper

text: bound

text: on

text: z

text: ,P

text: that

text: is,

text: surprisingly,

text: independent

text: zI

text: of

text: the

text: data

text: A,

text: b,

text: c,

text: and

text: depends

text: only

text: on

text: the

text: packing

text: interaction

text: graph

text: GpaJck

text: and

text: the

text: choice

text: of

text: V

text: A,

text: sparse

text: cut

text: support

text: list

text: J

text: Theorem

text: Consider

text: a

text: packing

text: integer

text: program

text: as

text: deﬁned

text: in

text: (P).

text: Let

text: be

text: a

text: partition

text: of

text: the

text: index

text: set

text: of

text: columns

text: of

text: A

text: and

text: let

text: G

text: GpacJk

text: (V,

text: E)

text: be

text: the

text: packing

text: interaction

text: graph

text: of

text: V

text: A,

text: A.

text: Then

text: for

text: any

text: sparse

text: cut

text: support

text: list

text: we

text: have

text: V

text: V

text: z

text: ,P

text: η

text: (G)

text: zI.

text: V

text: As

text: discussed

text: before,

text: if

text: we

text: are

text: considering

text: the

text: super

text: sparse

text: closure

text: P

text: S.S.,

text: η

text: (G)

text: is

text: the

text: usual

text: fractional

text: chromatic

text: number.

text: Therefore

text: we

text: obtain

text: the

text: following

text: possibly

text: weaker

text: bound

text: using

text: Brook’s

text: theorem

text: J

text: Corollary

text: Consider

text: a

text: packing

text: integer

text: program

text: as

text: deﬁned

text: in

text: (P).

text: Let

text: be

text: a

text: partition

text: of

text: the

text: index

text: set

text: of

text: columns

text: of

text: A

text: and

text: let

text: GpacJk

text: be

text: the

text: packing

text: interaction

text: graph

text: of

text: A.

text: A,

text: Let

text: denote

text: the

text: maximum

text: degree

text: of

text: G.

text: Then

text: we

text: have

text: the

text: following

text: bounds

text: on

text: the

text: optimum

text: value

text: of

text: the

text: super

text: sparse

text: closure

text: P

text: S.S:

text: If

text: GpacJk

text: is

text: not

text: a

text: complete

text: graph

text: or

text: an

text: odd

text: cycle,

text: then

text: A,

text: zS.S.

text: zI.

text: ##LTLine##


text: { } Remark. We strongly suspect that R( 1, eiπ/5, eiπ/4, eiπ/3 ) is not a ring, so we suspect that it is not suﬃcient for U to just be a subset of a ﬁnite group. { } Theorem 4.1. Let U = 1, α, α2, α3 . R(U) is a ring. Proof. Set z = I (0, 1), z = I (0, 1), and z = I (0, 1). Since the 1 α,α3 2 α,α2 − 3 − α2,α−3 only elementary monomials are 0, 1, z , z , z , 1 z , 1 z , 1 z , it suﬃces to { }1 2 3 1 2 3 check pairwise products of z , z , z . 1 2 3 Z Set p = I (0, z ), p = I (0, z ), and p = I (0, z ). Then [P] = Z 1 1,α2 1 2 1,α3 2 3 − 1,α − 3 − [p , p , p ], since the other projections are 0, 1, 1 p , 1 p , and 1 p . 1 2 3 1 2 2 First we claim that z z = z . We will prove this by calculation. 1 2 3 [1, α3] [1, α2] z z = α2 1 2 [α, α3] [α, α2] − − − − e 3iθ e3iθ e 2iθ e2iθ = − − − − e 2iθ e2iθ e iθ eiθ [1, α3] [1, α3] = α2 = α2 [α, α2] [α2, α3] = z 3 Next we claim that z /z = p and z /z = p . These can also be calculated 1 2 1 2 1 2 but a geometrical ﬁgure makes it clear. − − − − The ﬁrst claim follows from the fact that the triangles 0 p z and 0 1 z 1 1 − − 2 are similar. The second claim follows from the similarity of the triangles 0 1 z − − 1 and 0 p z . 2 2 z 3 z 2 z 1 p3 0 p1 1 p2 So far we can construct the following pairwise products of elementary mono- mials. z z2 = z z 1 = p z 1 1 2 z 1 3 2 z z = z 1 2 3 z2 = p2p z = p z 2 2 1 3 2 3 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Remark.

text: We

text: strongly

text: suspect

text: that

text: R(

text: is

text: not

text: a

text: ring,

text: so

text: we

text: suspect

text: that

text: it

text: is

text: not

text: suﬃcient

text: for

text: U

text: to

text: just

text: be

text: a

text: subset

text: of

text: a

text: ﬁnite

text: group.

text: Theorem

text: Let

text: U

text: α,

text: α3

text: R(U)

text: is

text: a

text: ring.

text: Proof.

text: Set

text: z

text: I

text: z

text: I

text: and

text: z

text: I

text: Since

text: the

text: α,α3

text: α,α2

text: α2,α−3

text: only

text: elementary

text: monomials

text: are

text: z

text: z

text: z

text: z

text: z

text: z

text: it

text: suﬃces

text: to

text: check

text: pairwise

text: products

text: of

text: z

text: z

text: z

text: Z

text: Set

text: p

text: I

text: z

text: p

text: I

text: z

text: and

text: p

text: I

text: z

text: Then

text: [P]

text: Z

text: [p

text: p

text: p

text: since

text: the

text: other

text: projections

text: are

text: p

text: p

text: and

text: p

text: First

text: we

text: claim

text: that

text: z

text: z

text: z

text: We

text: will

text: prove

text: this

text: by

text: calculation.

text: z

text: z

text: α2

text: [α,

text: [α,

text: e

text: e

text: e

text: e

text: iθ

text: eiθ

text: α2

text: α2

text: [α,

text: z

text: Next

text: we

text: claim

text: that

text: z

text: /z

text: p

text: and

text: z

text: /z

text: p

text: These

text: can

text: also

text: be

text: calculated

text: but

text: a

text: geometrical

text: ﬁgure

text: makes

text: it

text: clear.

text: The

text: ﬁrst

text: claim

text: follows

text: from

text: the

text: fact

text: that

text: the

text: triangles

text: p

text: z

text: and

text: z

text: are

text: similar.

text: The

text: second

text: claim

text: follows

text: from

text: the

text: similarity

text: of

text: the

text: triangles

text: z

text: and

text: p

text: z

text: z

text: z

text: z

text: So

text: far

text: we

text: can

text: construct

text: the

text: following

text: pairwise

text: products

text: of

text: elementary

text: mono-

text: mials.

text: z

text: z

text: z

text: p

text: z

text: z

text: z

text: z

text: z

text: z

text: p

text: z

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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