text: selcyc gnilaennA )s( emit revlos lacissalC 7 to limit these early investigations to the most relevant a) 105 b) 10-1 instances, and to lay out empirical procedures that work under some general assumptions. More speciﬁcally, we 104 focus on solving the CSP version of JSP, not the full 10-2 optimization problem, and we therefore only benchmark 1000 the Hamiltonians with T with the D-Wave machine. = T 10-3 We note however that a full optimization solver can be 100 realized by leveraging data analysis of past results on parametrized ensembles and by implementing an adap- 10 10-4 tive binary search. Full details can be found in a longer version of this work [31]. 1 10-5 On the quantum annealer installed at NASA Ames (it 0 100 200 300 400 10-5 10-4 10-3 10-2 10-1 has 509 working qubits; details are presented in [32]), Number of qubits D-Wave machine annealing time (s) we run hundreds of instances, sampling the ensembles ∈ { }, ∈ { }, and ∈ { }. FIG. 4: a) Number of repetitions required to solve HT with N = M 3, 4, 5, 6 θ 0.5, 1 Pp [1, 1], [0, 2] the D-Wave Two with a 99% probability of success (see the For each instance, we report results, such as runtimes, appendix). The blue points indicate instances with θ = 1 at the most optimal among those tested, assuming JF and yellow points correspond to θ = 0.5 (they are the same the application of an optimized parameter-setting pro- instances and runtimes used for Figure 3-c). The number of cedure along the lines of that described in the previous qubits on the x-axis represents the qubits used after embed- section. Figure 4-a displays the total annealing repeti- ding. b) Correlation plot between classical solvers and the tions required to achieve a 99% probability of success D-Wave optimizer. Gray and violet points represent runtimes on the ground state of , with each repetition lasting compared with algorithm B, and cyan and red are compared HT µ , as a function of the number of qubits in the to the MS algorithm, respectively, with θ = 1 and θ = 0.5. t = 20 s eAmbedded (and pruned) Hamiltonian. We observe an ex- All results presented correspond to the best out of 5 gauges selected randomly for every instance. In case the machine ponential increase in complexity with increasing Hamil- returns embedding components whose values are discordant, tonian size, for both classes of problems studied. This we apply a majority voting rule to recover a solution within likely means that while the problems tested are small, the the logical subspace [6, 11, 28, 33, 34]. We observe a devi- analog optimization procedure intrinsic to the D-Wave ation of about an order of magnitude on the annealing time device’s operation is already subject to the fundamental if we average over 5 gauges instead of picking the best one, complexity bottlenecks of the JSP. It is, however, pre- indicating that there is considerable room for improvement if mature to draw conclusions about performance scaling we were to apply more-advanced calibration techniques [32]. of the technology given the current constraints on cali- bration procedures, annealing time, etc. Many of these problems are expected to be either overcome or nearly so ticularly hard families of problems, not unlike what is with the next generation of D-Wave chip, at which point observed for the quantum annealing implementation of more extensive experimentation will be warranted. planning problems based on graph vertex coloring [9]. In Figure 4-b, we compare the performance of the The second algorithm, MS, introduced in [20], proposes D-Wave device to two exhaustive classical algorithms in a time-based branching scheme where a decision is made order to gain insight on how current quantum anneal- at each node to either schedule or delay one of the avail- ing technology compares with paradigmatic classical op- able operations at the current time. The authors then timization methods. Leaving the performance of approx- rely on a series of shaving procedures such as those pro- imate solutions for future work, we chose not to explore posed by [21] to determine the new bound and whether the plethora of possible heuristic methods as we operate the choice leads to valid schedules. This algorithm is a the D-Wave machine, seeking the global optimum. natural comparison with the present quantum annealing The ﬁrst algorithm, B, detailed in [35], exploits the dis- approach as it solves the decision version of the JSP in a junctive graph representation and a branch-and-bound very similar fashion to the time-indexed formulation we strategy that very eﬀectively combines a branching have implemented on the D-Wave machine, and it makes scheme based on selecting the direction of a single dis- use of the same shaving technique that we adapted as junctive edge (according to some single-machine con- a pre-processing step for variable pruning. However, we straints), and a technique introduced in [36] for ﬁxing should mention that the variable pruning that we im- additional disjunctions (based on a preemptive relax- plemented to simplify H is employed at each node of T ation). It has publicly available code and is considered the classical branch and bound algorithm, so the overall a well-performing complete solver for the small instances computational time of MS is usually much more important currently accessible to us, while remaining competitive than our one-pass pre-processing step, and in general its for larger ones even if other classical approaches become runtime does not scale polynomially with the problem more favorable [37]. B has been used in [38] to discuss size. the possibility of a phase transition in the JSP, demon- What is apparent from the correlation plot in strating that the random instances with are par- Figure 4-b is that the D-Wave machine is easily outper- N = M ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: the

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text: optimization

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text: and

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text: D-Wave

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text: T

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text: note

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text: For

text: each

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text: we

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text: results,

text: such

text: as

text: runtimes,

text: appendix).

text: The

text: blue

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text: number

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text: the

text: lines

text: of

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text: in

text: the

text: previous

text: qubits

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text: the

text: x-axis

text: represents

text: the

text: qubits

text: used

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text: section.

text: Figure

text: displays

text: the

text: total

text: annealing

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text: b)

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text: plot

text: between

text: classical

text: solvers

text: and

text: the

text: tions

text: required

text: to

text: achieve

text: a

text: probability

text: of

text: success

text: D-Wave

text: optimizer.

text: Gray

text: and

text: violet

text: points

text: represent

text: runtimes

text: on

text: the

text: ground

text: state

text: of

text: with

text: each

text: repetition

text: lasting

text: compared

text: with

text: algorithm

text: B,

text: and

text: cyan

text: and

text: red

text: are

text: compared

text: HT

text: µ

text: as

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text: function

text: of

text: the

text: number

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text: qubits

text: in

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text: to

text: the

text: MS

text: algorithm,

text: respectively,

text: with

text: θ

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text: θ

text: t

text: s

text: eAmbedded

text: (and

text: pruned)

text: Hamiltonian.

text: We

text: observe

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text: All

text: results

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text: to

text: the

text: best

text: out

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text: gauges

text: selected

text: randomly

text: for

text: every

text: instance.

text: In

text: case

text: the

text: machine

text: ponential

text: increase

text: in

text: complexity

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text: increasing

text: Hamil-

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text: embedding

text: components

text: whose

text: values

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text: discordant,

text: tonian

text: size,

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text: both

text: classes

text: of

text: problems

text: studied.

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text: we

text: apply

text: a

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text: rule

text: to

text: recover

text: a

text: solution

text: within

text: likely

text: means

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text: while

text: the

text: problems

text: tested

text: are

text: small,

text: the

text: the

text: logical

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text: observe

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text: analog

text: optimization

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text: to

text: the

text: D-Wave

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text: of

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text: an

text: order

text: of

text: magnitude

text: on

text: the

text: annealing

text: time

text: device’s

text: operation

text: is

text: already

text: subject

text: to

text: the

text: fundamental

text: if

text: we

text: average

text: over

text: gauges

text: instead

text: of

text: picking

text: the

text: best

text: one,

text: complexity

text: bottlenecks

text: of

text: the

text: JSP.

text: It

text: is,

text: however,

text: pre-

text: indicating

text: that

text: there

text: is

text: considerable

text: room

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text: mature

text: to

text: draw

text: conclusions

text: about

text: performance

text: scaling

text: we

text: were

text: to

text: apply

text: more-advanced

text: calibration

text: techniques

text: of

text: the

text: technology

text: given

text: the

text: current

text: constraints

text: on

text: cali-

text: bration

text: procedures,

text: annealing

text: time,

text: etc.

text: Many

text: of

text: these

text: problems

text: are

text: expected

text: to

text: be

text: either

text: overcome

text: or

text: nearly

text: so

text: ticularly

text: hard

text: families

text: of

text: problems,

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text: unlike

text: what

text: is

text: with

text: the

text: next

text: generation

text: of

text: D-Wave

text: chip,

text: at

text: which

text: point

text: observed

text: for

text: the

text: quantum

text: annealing

text: implementation

text: of

text: more

text: extensive

text: experimentation

text: will

text: be

text: warranted.

text: planning

text: problems

text: based

text: on

text: graph

text: vertex

text: coloring

text: In

text: Figure

text: we

text: compare

text: the

text: performance

text: of

text: the

text: The

text: second

text: algorithm,

text: MS,

text: introduced

text: in

text: proposes

text: D-Wave

text: device

text: to

text: two

text: exhaustive

text: classical

text: algorithms

text: in

text: a

text: time-based

text: branching

text: scheme

text: where

text: a

text: decision

text: is

text: made

text: order

text: to

text: gain

text: insight

text: on

text: how

text: current

text: quantum

text: anneal-

text: at

text: each

text: node

text: to

text: either

text: schedule

text: or

text: delay

text: one

text: of

text: the

text: avail-

text: ing

text: technology

text: compares

text: with

text: paradigmatic

text: classical

text: op-

text: able

text: operations

text: at

text: the

text: current

text: time.

text: The

text: authors

text: then

text: timization

text: methods.

text: Leaving

text: the

text: performance

text: of

text: approx-

text: rely

text: on

text: a

text: series

text: of

text: shaving

text: procedures

text: such

text: as

text: those

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text: imate

text: solutions

text: for

text: future

text: work,

text: we

text: chose

text: not

text: to

text: explore

text: posed

text: by

text: to

text: determine

text: the

text: new

text: bound

text: and

text: whether

text: the

text: plethora

text: of

text: possible

text: heuristic

text: methods

text: as

text: we

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text: the

text: choice

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text: to

text: valid

text: schedules.

text: This

text: algorithm

text: is

text: a

text: the

text: D-Wave

text: machine,

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text: the

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text: natural

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text: with

text: the

text: present

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text: annealing

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text: ﬁrst

text: algorithm,

text: B,

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text: in

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text: the

text: dis-

text: approach

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text: the

text: decision

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text: JSP

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text: a

text: branch-and-bound

text: very

text: similar

text: fashion

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text: the

text: time-indexed

text: formulation

text: we

text: strategy

text: that

text: very

text: eﬀectively

text: combines

text: a

text: branching

text: have

text: implemented

text: on

text: the

text: D-Wave

text: machine,

text: and

text: it

text: makes

text: scheme

text: based

text: on

text: selecting

text: the

text: direction

text: of

text: a

text: single

text: dis-

text: use

text: of

text: the

text: same

text: shaving

text: technique

text: that

text: we

text: adapted

text: as

text: junctive

text: edge

text: (according

text: to

text: some

text: single-machine

text: con-

text: a

text: pre-processing

text: step

text: for

text: variable

text: pruning.

text: However,

text: we

text: straints),

text: and

text: a

text: technique

text: introduced

text: in

text: for

text: ﬁxing

text: should

text: mention

text: that

text: the

text: variable

text: pruning

text: that

text: we

text: im-

text: additional

text: disjunctions

text: (based

text: on

text: a

text: preemptive

text: relax-

text: plemented

text: to

text: simplify

text: H

text: is

text: employed

text: at

text: each

text: node

text: of

text: T

text: ation).

text: It

text: has

text: publicly

text: available

text: code

text: and

text: is

text: considered

text: the

text: classical

text: branch

text: and

text: bound

text: algorithm,

text: so

text: the

text: overall

text: a

text: well-performing

text: complete

text: solver

text: for

text: the

text: small

text: instances

text: computational

text: time

text: of

text: MS

text: is

text: usually

text: much

text: more

text: important

text: currently

text: accessible

text: to

text: us,

text: while

text: remaining

text: competitive

text: than

text: our

text: one-pass

text: pre-processing

text: step,

text: and

text: in

text: general

text: its

text: for

text: larger

text: ones

text: even

text: if

text: other

text: classical

text: approaches

text: become

text: runtime

text: does

text: not

text: scale

text: polynomially

text: with

text: the

text: problem

text: more

text: favorable

text: B

text: has

text: been

text: used

text: in

text: to

text: discuss

text: size.

text: the

text: possibility

text: of

text: a

text: phase

text: transition

text: in

text: the

text: JSP,

text: demon-

text: What

text: is

text: apparent

text: from

text: the

text: correlation

text: plot

text: in

text: strating

text: that

text: the

text: random

text: instances

text: with

text: are

text: par-

text: Figure

text: is

text: that

text: the

text: D-Wave

text: machine

text: is

text: easily

text: outper-

text: N

text: M

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text: 9 Mitacs, NASA (Sponsor Award Number NNX12AK33A), . The following lemma summarizes the equivalent con- 0 and by the Oﬃce of the Director of National Intelligence dition for the reward-based case. (ODNI), Intelligence Advanced Research Projects Activ- Lemma 1. If β/η(cid:48) ≥ 3 and α > 0, then ity (IARPA), via IAA 145483 and by the AFRL Infor- mation Directorate under grant F4HBKC4162G001. (cid:48) ≥ − − (A3) Appendix A: JSP and QUBO formulation H (x¯) (k N), T N In this appendix we expand on the penalty form used for all , with equality if and only if represents a for the constraints and the alternative reward-base for- x¯ x¯ feasible schedule. mulation as well as the timespan discrimination scheme. 1. Penalties versus rewards formulation We also found examples that show that these bounds on the coeﬃcients are tight. The encoding of constraints as terms in a QUBO prob- The fact that (cid:48) must be greater than or equal to 3 is β/η lem can either reward the respecting of these constraints a clear disadvantage because of the issues with precision or penalize their violation. Although the distinction may of the current hardware. In we can set all of the H at ﬁrst seem artiﬁcial, the actual QUBO problem gener- penalty coeﬃcients (and hence aTll non-zero couplers) to ated diﬀers and can lead to diﬀerent performance on an be equal, which is the best possible case from the point imperfect annealer. We present one such alternative for- of view of precision. mulation where the precedence constraint (7) is instead encoded as a reward for correct ordering by replacing with − (cid:48) (cid:48) , where +ηh (x¯) η h (x¯)  2. Timespan discrimination 1 1 (cid:88)  (cid:88)    The timespan discrimination that we propose is a spec-   iﬁcation to strike a compromise between the information (cid:48) (A1) obtained from each solver call, and the required preci- h (x¯) = x x (cid:48) . 1 i,t i+1,t sion for this information to be accurate and obtained n kn−1<≤i<(cid:48)kn eﬃciently. Speciﬁcally, we want this extra information t+pi t to help identify the optimal makespan by looking at the The new Hamiltonian is energy of the solutions. This means breaking the degen- eracy of the ground states (i.e., the valid solutions) and assigning diﬀerent energy sectors to diﬀerent makespans. H(cid:48) (x¯) = −η(cid:48)h(cid:48) (x¯) + αh (x¯) + βh (x¯). (A2) To prevent collisions with invalid solutions, these energy T 1 2 3 sectors have to ﬁt within the logical QUBO problem’s gap given by { }. We note that this will The reward attributed to a solution is equal to (cid:48) times ∆E = min η, α, β η aﬀect the actual gap (as seen by the hardware) of the the number of satisﬁed precedence constraints. A feasi- embedded Ising model. ble solution, where all constraints are satisﬁed, will have Since the binary variables we have deﬁned in the pro- energy equal to − (cid:48) − . η (k N) posed formulation are not suﬃcient to write a simple ex- The functions anNd (cid:48) diﬀer only by the range of (cid:48): h h t pression for the makespan of a given solution, additional in the rewards ver1sion we1have auxiliary variables and associated constraints would need (cid:48) − ≥ to be introduced. Instead, a simple way to implement t t pi, this feature in our QUBO formulation is to add a num- ber of local ﬁelds to the binary variables corresponding and in the penalties version we have to the last operation of each job, { }. Since O , . . . , O (cid:48) − the makespan depends on the complekti1on timekoNf the last t t < p . i operation, the precedence constraint guaranties that the The fact that we are allowing equality in the rewards makespan of a valid solution will be equal to the com- version means that (cid:48) will always have more quadratic pletion time of one of those operations. We can then h terms than regardl1ess of variable pruning, leading to select the local ﬁeld appropriately as a function of the h 1 a more connected QUBO graph and therefore a harder time index to penalize a ﬁxed number of the larger t K problem to embed. makespans ranging from − to . Within a sector T K +1 T Another important disadvantage is revealed when assigned to the time step T , we need to further divide choosing the coeﬃcients (cid:48), , and in (cid:48) to guar- by the maximum number of operations that can η α β H ∆ET antee that no unfeasible solution has energyT less than complete at T to obtain the largest value we can use as − (cid:48) − . This can happen if the penalty that we the local ﬁeld , i.e., the number of distinct machines η (k N) hT N gain from breaking constraints or is less than the used by at least one operation in the set of operations h h potential reward we get from (cid:48) . 2The p3enalty-based for- { }, denoted by . If is larger than mulation simply requires thath1, , and be larger than 1,Owk1e,a. l.s.o, OnekeNd to ensure thatMcoﬁnnatrl ibutiKons from various η α β

label: paragraph

text: Mitacs,

text: NASA

text: (Sponsor

text: Award

text: Number

text: The

text: following

text: lemma

text: summarizes

text: the

text: equivalent

text: con-

text: and

text: by

text: the

text: Oﬃce

text: of

text: the

text: Director

text: of

text: National

text: Intelligence

text: dition

text: for

text: the

text: reward-based

text: case.

text: (ODNI),

text: Intelligence

text: Advanced

text: Research

text: Projects

text: Activ-

text: Lemma

text: If

text: and

text: α

text: then

text: ity

text: (IARPA),

text: via

text: IAA

text: and

text: by

text: the

text: AFRL

text: Infor-

text: mation

text: Directorate

text: under

text: grant

label: equation

text: Appendix

label: section

text: A:

text: JSP

text: and

text: QUBO

text: formulation

text: H

text: (x¯)

text: (k

text: N),

text: T

text: N

text: In

text: this

text: appendix

text: we

text: expand

text: on

text: the

text: penalty

text: form

text: used

text: for

text: all

text: with

text: equality

text: if

text: and

text: only

text: if

text: represents

text: a

text: for

text: the

text: constraints

text: and

text: the

text: alternative

text: reward-base

text: for-

text: x¯

text: x¯

text: feasible

text: schedule.

text: mulation

text: as

text: well

text: as

text: the

text: timespan

text: discrimination

text: scheme.

text: Penalties

text: versus

text: rewards

text: formulation

text: We

text: also

text: found

text: examples

text: that

text: show

text: that

text: these

text: bounds

text: on

text: the

text: coeﬃcients

text: are

text: tight.

text: The

text: encoding

text: of

text: constraints

text: as

text: terms

text: in

text: a

text: QUBO

text: prob-

text: The

text: fact

text: that

text: must

text: be

text: greater

text: than

text: or

text: equal

text: to

text: is

text: β/η

text: lem

text: can

text: either

text: reward

text: the

text: respecting

text: of

text: these

text: constraints

text: a

text: clear

text: disadvantage

text: because

text: of

text: the

text: issues

text: with

text: precision

text: or

text: penalize

text: their

text: violation.

text: Although

text: the

text: distinction

text: may

text: of

text: the

text: current

text: hardware.

text: In

text: we

text: can

text: set

text: all

text: of

text: the

text: H

text: at

text: ﬁrst

text: seem

text: artiﬁcial,

text: the

text: actual

text: QUBO

text: problem

text: gener-

text: penalty

text: coeﬃcients

text: (and

text: hence

text: aTll

text: non-zero

text: couplers)

text: to

text: ated

text: diﬀers

text: and

text: can

text: lead

text: to

text: diﬀerent

text: performance

text: on

text: an

text: be

text: equal,

text: which

text: is

text: the

text: best

text: possible

text: case

text: from

text: the

text: point

text: imperfect

text: annealer.

text: We

text: present

text: one

text: such

text: alternative

text: for-

text: of

text: view

text: of

text: precision.

text: mulation

text: where

text: the

text: precedence

text: constraint

text: is

text: instead

text: encoded

text: as

text: a

text: reward

text: for

text: correct

text: ordering

text: by

text: replacing

text: with

text: where

text: +ηh

text: (x¯)

text: η

text: h

text: (x¯)

text: Timespan

text: discrimination

text: The

text: timespan

text: discrimination

text: that

text: we

text: propose

text: is

text: a

text: spec-

text: iﬁcation

text: to

text: strike

text: a

text: compromise

text: between

text: the

text: information

text: obtained

text: from

text: each

text: solver

text: call,

text: and

text: the

text: required

text: preci-

text: h

text: (x¯)

text: x

text: x

text: i,t

text: sion

text: for

text: this

text: information

text: to

text: be

text: accurate

text: and

text: obtained

text: n

text: eﬃciently.

text: Speciﬁcally,

text: we

text: want

text: this

text: extra

text: information

text: t+pi

text: t

text: to

text: help

text: identify

text: the

text: optimal

text: makespan

text: by

text: looking

text: at

text: the

text: The

text: new

text: Hamiltonian

text: is

text: energy

text: of

text: the

text: solutions.

text: This

text: means

text: breaking

text: the

text: degen-

text: eracy

text: of

text: the

text: ground

text: states

text: (i.e.,

text: the

text: valid

text: solutions)

text: and

text: assigning

text: diﬀerent

text: energy

text: sectors

text: to

text: diﬀerent

text: makespans.

text: (x¯)

text: (x¯)

text: αh

text: (x¯)

text: βh

text: (x¯).

text: To

text: prevent

text: collisions

text: with

text: invalid

text: solutions,

text: these

text: energy

text: T

text: sectors

text: have

text: to

text: ﬁt

text: within

text: the

text: logical

text: QUBO

text: problem’s

text: gap

text: given

text: by

text: We

text: note

text: that

text: this

text: will

text: The

text: reward

text: attributed

text: to

text: a

text: solution

text: is

text: equal

text: to

text: times

text: ∆E

text: min

text: η,

text: α,

text: β

text: η

text: aﬀect

text: the

text: actual

text: gap

text: (as

text: seen

text: by

text: the

text: hardware)

text: of

text: the

text: the

text: number

text: of

text: satisﬁed

text: precedence

text: constraints.

text: A

text: feasi-

text: embedded

text: Ising

text: model.

text: ble

text: solution,

text: where

text: all

text: constraints

text: are

text: satisﬁed,

text: will

text: have

text: Since

text: the

text: binary

text: variables

text: we

text: have

text: deﬁned

text: in

text: the

text: pro-

text: energy

text: equal

text: to

text: η

text: (k

text: N)

text: posed

text: formulation

text: are

text: not

text: suﬃcient

text: to

text: write

text: a

text: simple

text: ex-

text: The

text: functions

text: anNd

text: diﬀer

text: only

text: by

text: the

text: range

text: of

text: h

text: h

text: t

text: pression

text: for

text: the

text: makespan

text: of

text: a

text: given

text: solution,

text: additional

text: in

text: the

text: rewards

text: auxiliary

text: variables

text: and

text: associated

text: constraints

text: would

text: need

text: to

text: be

text: introduced.

text: Instead,

text: a

text: simple

text: way

text: to

text: implement

text: t

text: t

text: pi,

text: this

text: feature

text: in

text: our

text: QUBO

text: formulation

text: is

text: to

text: add

text: a

text: num-

text: ber

text: of

text: local

text: ﬁelds

text: to

text: the

text: binary

text: variables

text: corresponding

text: and

text: in

text: the

text: penalties

text: version

text: we

text: have

text: to

text: the

text: last

text: operation

text: of

text: each

text: job,

text: Since

text: O

text: O

text: the

text: makespan

text: depends

text: on

text: the

text: timekoNf

text: the

text: last

text: t

text: t

text: p

text: i

text: operation,

text: the

text: precedence

text: constraint

text: guaranties

text: that

text: the

text: The

text: fact

text: that

text: we

text: are

text: allowing

text: equality

text: in

text: the

text: rewards

text: makespan

text: of

text: a

text: valid

text: solution

text: will

text: be

text: equal

text: to

text: the

text: com-

text: version

text: means

text: that

text: will

text: always

text: have

text: more

text: quadratic

text: pletion

text: time

text: of

text: one

text: of

text: those

text: operations.

text: We

text: can

text: then

text: h

text: terms

text: than

text: of

text: variable

text: pruning,

text: leading

text: to

text: select

text: the

text: local

text: ﬁeld

text: appropriately

text: as

text: a

text: function

text: of

text: the

text: h

text: a

text: more

text: connected

text: QUBO

text: graph

text: and

text: therefore

text: a

text: harder

text: time

text: index

text: to

text: penalize

text: a

text: ﬁxed

text: number

text: of

text: the

text: larger

text: t

text: K

text: problem

text: to

text: embed.

text: makespans

text: ranging

text: from

text: to

text: Within

text: a

text: sector

text: T

text: K

text: T

text: Another

text: important

text: disadvantage

text: is

text: revealed

text: when

text: assigned

text: to

text: the

text: time

text: step

text: T

text: we

text: need

text: to

text: further

text: divide

text: choosing

text: the

text: coeﬃcients

text: and

text: in

text: to

text: guar-

text: by

text: the

text: maximum

text: number

text: of

text: operations

text: that

text: can

text: η

text: α

text: β

text: H

text: ∆ET

text: antee

text: that

text: no

text: unfeasible

text: solution

text: has

text: energyT

text: less

text: than

text: complete

text: at

text: T

text: to

text: obtain

text: the

text: largest

text: value

text: we

text: can

text: use

text: as

text: This

text: can

text: happen

text: if

text: the

text: penalty

text: that

text: we

text: the

text: local

text: ﬁeld

text: i.e.,

text: the

text: number

text: of

text: distinct

text: machines

text: η

text: (k

text: N)

text: hT

text: N

text: gain

text: from

text: breaking

text: constraints

text: or

text: is

text: less

text: than

text: the

text: used

text: by

text: at

text: least

text: one

text: operation

text: in

text: the

text: set

text: of

text: operations

text: h

text: h

text: potential

text: reward

text: we

text: get

text: from

text: for-

text: denoted

text: by

text: If

text: is

text: larger

text: than

text: mulation

text: simply

text: requires

text: and

text: be

text: larger

text: than

text: l.s.o,

text: OnekeNd

text: to

text: ensure

text: thatMcoﬁnnatrl

text: ibutiKons

text: from

text: various

text: η

text: α

text: β


text: [10] Kr´al D., Kratochv´ıl J., Tuza Z., Woeginger G.: Complexity of Coloring Graphs without For- bidden Induced Subgraphs. Graph-Theoretic Concepts in Computer Science, 27th International Workshop, WG 2001, Boltenhagen, Germany, June 14-16, 2001, Proceedings. (2001) 254–262 [11] Lokshtanov D., Vatshelle M., Villanger Y.: Independent Set in P -Free Graphs in Polynomial 5 Time submitted for publication. (2013) [12] Neumann-Lara V.: The dichromatic number of a digraph. J. Comb. Theory, Ser. B 33 (1982) 265–270 [13] Randerath B., Schiermeyer I.: 3-Colorability in P for P -free graphs. Discrete Applied Mathe- 6 matics. 136 (2004) 299–313 [14] Robertson N., Seymour P.: Graph Minors. XX. Wagner’s conjecture. J. Comb. Theory, Ser. B. 92 (2004) 325–357 [15] Woeginger G., Sgall J.: The complexity of coloring graphs without long induced paths. Acta Cybern. 15 (2001) 107–117 [16] Choromanski K.: The strong EH-property and the Erd˝os-Hajnal conjecture, submitted for pub- lication. [17] Choromanski K.: All known prime Erd˝os-Hajnal tournaments satisfy (cid:15)(H) = Ω(| | 1 | | ), H 5 log( H ) submitted for publication. [18] Choromanski K., Jebara T.: Coloring tournaments with forbidden substructures, submitted for publication. [19] Berger E., Choromanski K., Chudnovsky M., Fox J., Loebl M., Scott A., Seymour P., Thomass´e S.: Tournaments and coloring. Journal of Combinatorial Theory, Ser. B 103 (2013), p.1-20. [20] Choromanski K., Chudnovsky M., Seymour P.: Tournaments with near-linear transitive subsets, Journal of Combinatorial Theory, Ser. B (2014). [21] Choromanski K.: Excluding pairs of tournaments, submitted for publication. [22] Choromanski K.: Upper Bounds for Erd¨os-Hajnal Coeﬃcients of Tournaments, Journal of Graph Theory (2013). [23] Choromanski K.: Tournaments with forbidden substructures and the Erd¨os-Hajnal Conjecture, Ph.D dissertation, Columbia University (2013). [24] Choromanski K.: Learning how to rank from heavily perturbed statistics-digraph clustering approach, submitted for publication. 15 ##LTLine##

label: reference

text: Kr´al

text: D.,

text: Kratochv´ıl

text: J.,

text: Tuza

text: Z.,

text: Woeginger

text: G.:

text: Complexity

text: of

text: Coloring

text: Graphs

text: without

text: For-

text: bidden

text: Induced

text: Subgraphs.

text: Graph-Theoretic

text: Concepts

text: in

text: Computer

text: Science,

text: International

text: Workshop,

text: WG

text: Boltenhagen,

text: Germany,

text: June

text: Proceedings.

text: Lokshtanov

text: D.,

text: Vatshelle

text: M.,

text: Villanger

text: Y.:

text: Independent

text: Set

text: in

text: P

text: -Free

text: Graphs

text: in

text: Polynomial

text: Time

text: submitted

text: for

text: publication.

text: Neumann-Lara

text: V.:

text: The

text: dichromatic

text: number

text: of

text: a

text: digraph.

text: J.

text: Comb.

text: Theory,

text: Ser.

text: B

text: Randerath

text: B.,

text: Schiermeyer

text: I.:

text: in

text: P

text: for

text: P

text: -free

text: graphs.

text: Discrete

text: Applied

text: Mathe-

text: matics.

text: Robertson

text: N.,

text: Seymour

text: P.:

text: Graph

text: Minors.

text: XX.

text: Wagner’s

text: conjecture.

text: J.

text: Comb.

text: Theory,

text: Ser.

text: B.

text: Woeginger

text: G.,

text: Sgall

text: J.:

text: The

text: complexity

text: of

text: coloring

text: graphs

text: without

text: long

text: induced

text: paths.

text: Acta

text: Cybern.

text: Choromanski

text: K.:

text: The

text: strong

text: EH-property

text: and

text: the

text: Erd˝os-Hajnal

text: conjecture,

text: submitted

text: for

text: pub-

text: lication.

text: Choromanski

text: K.:

text: All

text: known

text: prime

text: Erd˝os-Hajnal

text: tournaments

text: satisfy

text: Ω(|

text: H

text: log(

text: H

text: submitted

text: for

text: publication.

text: Choromanski

text: K.,

text: Jebara

text: T.:

text: Coloring

text: tournaments

text: with

text: forbidden

text: substructures,

text: submitted

text: for

text: publication.

text: Berger

text: E.,

text: Choromanski

text: K.,

text: Chudnovsky

text: M.,

text: Fox

text: J.,

text: Loebl

text: M.,

text: Scott

text: A.,

text: Seymour

text: P.,

text: Thomass´e

text: S.:

text: Tournaments

text: and

text: coloring.

text: Journal

text: of

text: Combinatorial

text: Theory,

text: Ser.

text: B

text: Choromanski

text: K.,

text: Chudnovsky

text: M.,

text: Seymour

text: P.:

text: Tournaments

text: with

text: near-linear

text: transitive

text: subsets,

text: Journal

text: of

text: Combinatorial

text: Theory,

text: Ser.

text: B

text: Choromanski

text: K.:

text: Excluding

text: pairs

text: of

text: tournaments,

text: submitted

text: for

text: publication.

text: Choromanski

text: K.:

text: Upper

text: Bounds

text: for

text: Erd¨os-Hajnal

text: Coeﬃcients

text: of

text: Tournaments,

text: Journal

text: of

text: Graph

text: Theory

text: Choromanski

text: K.:

text: Tournaments

text: with

text: forbidden

text: substructures

text: and

text: the

text: Erd¨os-Hajnal

text: Conjecture,

text: Ph.D

text: dissertation,

text: Columbia

text: University

text: Choromanski

text: K.:

text: Learning

text: how

text: to

text: rank

text: from

text: heavily

text: perturbed

text: statistics-digraph

text: clustering

text: approach,

text: submitted

text: for

text: publication.

text: ##LTLine##


text: the ﬁrst color class, removes it from the tournament, runs FindTrans on the remaining tournament to ﬁnd the second color class, etc. 4.1 Algorithm FindTrans Before giving a description of the algorithm FindTrans, we need to introduce a few more deﬁnition. ⊆ e(X,Y ) For a tournament T and two disjoint nonempty subsets X, Y V (T ) we denote d(X, Y ) = | || | , X Y where e(X, Y ) is the number of directed edges of T going from X to Y . The expression d(X, Y ) basically encodes directed density of edges from X to Y . Input: k > 1 and α-sequence θ = (A , ..., A ) of length k; 1 k Output: α-sequence θ ; s begin (cid:83) let λ = 1 and λ = 4λk2; 32k{4 ∈ k| | ≤ | | − } ∈ { } (cid:54) let C = v A : Nθ(j) A (1 2kλ ) for i, j 1, ..., h , i = j; i,j ← i\ v j k update: A A (cid:54) C ; i i j=i i,j output (A , ..., A ); 1 k end Algorithm 1: Algorithm MakeSmooth We say that a sequence (A , ..., A ) of pairwise disjoint subsets of V (T ) is a (c, λ)-α-sequence of 1 l length l if the following holds: • | | ≥ | | A c T for i = 1, ..., l and i • ≥ − ≤ ≤ d(A , A ) 1 λ for 1 i < j l. i j If the parameters c, λ of the (c, λ)-α-sequence are not important, we simply refer to it as an α-sequence. We say that a (c, λ)-α-sequence (A , ..., A ) of length l is smooth if the following strenghtening of 1 l the second condition from the deﬁnition above holds: • { } ≥ − ∈ ≤ ≤ d( x , A ) 1 λ for x A , 1 i < j l and, j i • { } ≥ − ∈ ≤ ≤ d(A , y ) 1 λ for y A , 1 i < j l. i j ∈ (cid:54) Given an α-sequence θ = (A , ..., A ), a vertex v A and j = i we denote by Nθ(j): 1 l i v • a set of all outneighbors of v from A if j > i and, j • a set of all inneighbors of v from A if j < i. j ∪ ∪ For an α-sequence θ = (A , ..., A ) we denote: V (θ) = A ... A . Let θ = (A , ..., A ) and θ = 1 l ⊗1 l 1 1 l 2 (B , ..., B ) be two disjoint α-sequences. We denote by θ θ the α-sequence (A , ..., A , B , ..., B ). 1 r ≤ | | 1 2 1 l 1 r For a set A and m A we denote by tr(A, m) the truncated version of A obtained by taking arbitrar- ily its m elements. For an α-sequence θ = (A , ..., A ) we denote: tr(θ, m) = (tr(A , m), ..., tr(A , m)). 1 l 1 l If the order of the given P -free tournament is too small, the algorithm FindTrans (Algorithm 2) k returns a trivial answer (and it is easy to see that this gives good asymptotics on the coeﬃcient (cid:15)). 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: ﬁrst

text: color

text: class,

text: removes

text: it

text: from

text: the

text: tournament,

text: runs

text: FindTrans

text: on

text: the

text: remaining

text: tournament

text: to

text: ﬁnd

text: the

text: second

text: color

text: class,

text: etc.

text: Algorithm

text: FindTrans

text: Before

text: giving

text: a

text: description

text: of

text: the

text: algorithm

text: FindTrans,

text: we

text: need

text: to

text: introduce

text: a

text: few

text: more

text: deﬁnition.

text: e(X,Y

text: For

text: a

text: tournament

text: T

text: and

text: two

text: disjoint

text: nonempty

text: subsets

text: X,

text: Y

text: V

text: (T

text: we

text: denote

text: d(X,

text: Y

text: X

text: Y

text: where

text: e(X,

text: Y

text: is

text: the

text: number

text: of

text: directed

text: edges

text: of

text: T

text: going

text: from

text: X

text: to

text: Y

text: The

text: expression

text: d(X,

text: Y

text: basically

text: encodes

text: directed

text: density

text: of

text: edges

text: from

text: X

text: to

text: Y

text: Input:

text: k

text: and

text: α-sequence

text: θ

text: (A

text: A

text: of

text: length

text: k;

text: k

text: Output:

text: α-sequence

text: θ

text: s

text: begin

text: let

text: λ

text: and

text: λ

text: k|

text: let

text: C

text: v

text: A

text: Nθ(j)

text: A

text: for

text: i,

text: j

text: h

text: i

text: j;

text: i,j

text: i\

text: v

text: j

text: k

text: update:

text: A

text: A

text: C

text: i

text: i

text: j=i

text: i,j

text: output

text: (A

text: A

text: k

text: end

text: Algorithm

text: Algorithm

text: MakeSmooth

text: We

text: say

text: that

text: a

text: sequence

text: (A

text: A

text: of

text: pairwise

text: disjoint

text: subsets

text: of

text: V

text: (T

text: is

text: a

text: (c,

text: λ)-α-sequence

text: of

text: l

text: length

text: l

text: if

text: the

text: following

text: holds:

text: A

text: c

text: T

text: for

text: i

text: l

text: and

text: i

text: d(A

text: A

text: λ

text: for

text: i

text: j

text: l.

text: i

text: j

text: If

text: the

text: parameters

text: c,

text: λ

text: of

text: the

text: (c,

text: λ)-α-sequence

text: are

text: not

text: important,

text: we

text: simply

text: refer

text: to

text: it

text: as

text: an

text: α-sequence.

text: We

text: say

text: that

text: a

text: (c,

text: λ)-α-sequence

text: (A

text: A

text: of

text: length

text: l

text: is

text: smooth

text: if

text: the

text: following

text: strenghtening

text: of

text: l

text: the

text: second

text: condition

text: from

text: the

text: deﬁnition

text: above

text: holds:

text: d(

text: x

text: A

text: λ

text: for

text: x

text: A

text: i

text: j

text: l

text: and,

text: j

text: i

text: d(A

text: y

text: λ

text: for

text: y

text: A

text: i

text: j

text: l.

text: i

text: j

text: Given

text: an

text: α-sequence

text: θ

text: (A

text: A

text: a

text: vertex

text: v

text: A

text: and

text: j

text: i

text: we

text: denote

text: by

text: Nθ(j):

text: l

text: i

text: v

text: a

text: set

text: of

text: all

text: outneighbors

text: of

text: v

text: from

text: A

text: if

text: j

text: i

text: and,

text: j

text: a

text: set

text: of

text: all

text: inneighbors

text: of

text: v

text: from

text: A

text: if

text: j

text: i.

text: j

text: For

text: an

text: α-sequence

text: θ

text: (A

text: A

text: we

text: denote:

text: V

text: (θ)

text: A

text: A

text: Let

text: θ

text: (A

text: A

text: and

text: θ

text: l

text: ⊗1

text: l

text: l

text: (B

text: B

text: be

text: two

text: disjoint

text: α-sequences.

text: We

text: denote

text: by

text: θ

text: θ

text: the

text: α-sequence

text: (A

text: A

text: B

text: B

text: r

text: l

text: r

text: For

text: a

text: set

text: A

text: and

text: m

text: A

text: we

text: denote

text: by

text: tr(A,

text: m)

text: the

text: truncated

text: version

text: of

text: A

text: obtained

text: by

text: taking

text: arbitrar-

text: ily

text: its

text: m

text: elements.

text: For

text: an

text: α-sequence

text: θ

text: (A

text: A

text: we

text: denote:

text: tr(θ,

text: m)

text: (tr(A

text: m),

text: tr(A

text: m)).

text: l

text: l

text: If

text: the

text: order

text: of

text: the

text: given

text: P

text: -free

text: tournament

text: is

text: too

text: small,

text: the

text: algorithm

text: FindTrans

text: (Algorithm

text: k

text: returns

text: a

text: trivial

text: answer

text: (and

text: it

text: is

text: easy

text: to

text: see

text: that

text: this

text: gives

text: good

text: asymptotics

text: on

text: the

text: coeﬃcient

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Input: k > 0 and P -free tournament T ; k | | c Output: transitive subset in V (T ) of order T klog(k)2 ; begin | | if T = 1 then output V (T ); end let c = 1(λk )log(k)+1, where: λ = 1 ; k | k| ≤k2 32k4 if 1 < T k then c k output any 2-element subset of V (T ); end run CreateSequence(k,T) to obtain an α-sequence θ of length k; run MakeSmooth(k, θ) to obtain a smooth α-sequence (A , ..., A ); ← 1 k initialize: θ (A , ..., A ); s 1 k let θ (i) denote the ith element of θ ; s s for i = 1, ..., k do 2 let u = left(i); let v = right(i); if there exists an edge e = (y, x) from θ (v) to θ (u) then (cid:48) ← (cid:48) ← s s let A θ (v), A θ (u) and (cid:48) ←v s∩ u ∩ s ∈ { } \ { } At θs(t) ←Nyθs(cid:48)(t) N(cid:48)xθs(t) for t 1, ..., k v, u ; update: θ (A , ..., A ); s 1 k else | run F indT rans(k, T A ) to obtain a transitive subset M ; | u 1 run F indT rans(k, T A ) to obtain a transitive subset M ; ∪ v 2 output M M ; 1 2 end end end Algorithm 2: Algorithm FindTrans 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Input:

text: k

text: and

text: P

text: -free

text: tournament

text: T

text: k

text: c

text: Output:

text: transitive

text: subset

text: in

text: V

text: (T

text: of

text: order

text: T

text: begin

text: if

text: T

text: then

text: output

text: V

text: (T

text: end

text: let

text: c

text: where:

text: λ

text: k

text: k|

text: if

text: T

text: k

text: then

text: c

text: k

text: output

text: any

text: subset

text: of

text: V

text: (T

text: end

text: run

text: CreateSequence(k,T)

text: to

text: obtain

text: an

text: α-sequence

text: θ

text: of

text: length

text: k;

text: run

text: MakeSmooth(k,

text: θ)

text: to

text: obtain

text: a

text: smooth

text: α-sequence

text: (A

text: A

text: k

text: initialize:

text: θ

text: (A

text: A

text: s

text: k

text: let

text: θ

text: (i)

text: denote

text: the

text: ith

text: element

text: of

text: θ

text: s

text: s

text: for

text: i

text: k

text: do

text: let

text: u

text: left(i);

text: let

text: v

text: right(i);

text: if

text: there

text: exists

text: an

text: edge

text: e

text: (y,

text: x)

text: from

text: θ

text: (v)

text: to

text: θ

text: (u)

text: then

text: s

text: s

text: let

text: A

text: θ

text: (v),

text: A

text: θ

text: (u)

text: and

text: ←v

text: s∩

text: u

text: s

text: At

text: θs(t)

text: for

text: t

text: k

text: v,

text: u

text: update:

text: θ

text: (A

text: A

text: s

text: k

text: else

text: run

text: F

text: indT

text: rans(k,

text: T

text: A

text: to

text: obtain

text: a

text: transitive

text: subset

text: M

text: u

text: run

text: F

text: indT

text: rans(k,

text: T

text: A

text: to

text: obtain

text: a

text: transitive

text: subset

text: M

text: v

text: output

text: M

text: M

text: end

text: end

text: end

text: Algorithm

text: Algorithm

text: FindTrans

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: ##LTLine##


text: Otherwise, the algorithm uses two subprocedures: CreateSequence that constructs in the P -free k tournament T an α-sequence of length k and MakeSmooth that uses that sequence to construct a smooth α-sequence of the same length. The CreateSequence procedure does not rely on the structural properties of the P -free tournaments, but just uses the fact that a tournament it operates on is H-free k for some k-vertex forbidden pattern H. We will discuss it in much detail later. The MakeSmooth procedure (Algorithm 1) is a standard method for getting rid of these vertices from the given α- sequence that have many less in/out-neighbors in some element of the α-sequence than the density condition would suggest. Input: r > 1 and P -free n-vertex tournament T ; k Output: an α-sequence of length r in T ; begin { } let V (P ) = h , ..., h ; k 1 k partition V (T ) arbitrarily into k equal-size sets: S , ..., S ; { } 1 k ⊆ run MakeDenseP air( S , ..., S , P , n) to get (X, Y ), where X, Y V (T ); 1 k k if r = 2 then output (X, Y ); end L ← ∅ R ← ∅ initialize: , ; | | | | let s = X and s = Y ; 1 | | ≥ 2 while X s1 do 2 | run CreateSequence(r , T X) to obtain an α-sequence L of length r ; ← \ 2 L ← L ∪ { } 2 update: X X V (L), L ; end | | ≥ while Y s2 do 2 | run CreateSequence(r , T Y ) to obtain an α-sequence R of length r ; ← \ 2 R ← R ∪ { } 2 update: Y Y V (R), R ; end − let λ = 321k4∈ cLr = (∈2c)Rlog(r) 1c and m = c2ncr2 , whe≥re c−= λkk2 ; if exists L , R such that d(V (L), V (R)) 1 4λ then ⊗ output tr(L R, m); else ⊗ ∈ L ∈ R output θ θ , for arbitrary: θ and θ ; 1 2 1 2 end end Algorithm 3: Algorithm CreateSequence Let us assume now that a smooth α-sequence of length k is given. The algorithm FindTrans tries to reconstruct a directed path P by looking for its ith vertex in the matching ordering of P in the k k ith element of the α-sequence θ . This is conducted backward edge by backward edge in the matching s ordering of P . If the backward edge is not found then two linear-size subsets A , A of two distinct k u v elements from the original α-sequence such that A is complete to A are detected. Otherwise an u v α-sequence is updated. The update is done in such a way that if in the new α-sequence the other backward edges of the matching ordering of P are found then they can be combined with the edges k 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Otherwise,

text: the

text: algorithm

text: uses

text: two

text: subprocedures:

text: CreateSequence

text: that

text: constructs

text: in

text: the

text: P

text: -free

text: k

text: tournament

text: T

text: an

text: α-sequence

text: of

text: length

text: k

text: and

text: MakeSmooth

text: that

text: uses

text: that

text: sequence

text: to

text: construct

text: a

text: smooth

text: α-sequence

text: of

text: the

text: same

text: length.

text: The

text: CreateSequence

text: procedure

text: does

text: not

text: rely

text: on

text: the

text: structural

text: properties

text: of

text: the

text: P

text: -free

text: tournaments,

text: but

text: just

text: uses

text: the

text: fact

text: that

text: a

text: tournament

text: it

text: operates

text: on

text: is

text: H-free

text: k

text: for

text: some

text: k-vertex

text: forbidden

text: pattern

text: H.

text: We

text: will

text: discuss

text: it

text: in

text: much

text: detail

text: later.

text: The

text: MakeSmooth

text: procedure

text: (Algorithm

text: is

text: a

text: standard

text: method

text: for

text: getting

text: rid

text: of

text: these

text: vertices

text: from

text: the

text: given

text: α-

text: sequence

text: that

text: have

text: many

text: less

text: in/out-neighbors

text: in

text: some

text: element

text: of

text: the

text: α-sequence

text: than

text: the

text: density

text: condition

text: would

text: suggest.

text: Input:

text: r

text: and

text: P

text: -free

text: n-vertex

text: tournament

text: T

text: k

text: Output:

text: an

text: α-sequence

text: of

text: length

text: r

text: in

text: T

text: begin

text: let

text: V

text: (P

text: h

text: h

text: k

text: k

text: partition

text: V

text: (T

text: arbitrarily

text: into

text: k

text: equal-size

text: sets:

text: S

text: S

text: k

text: run

text: MakeDenseP

text: air(

text: S

text: S

text: P

text: n)

text: to

text: get

text: (X,

text: Y

text: where

text: X,

text: Y

text: V

text: (T

text: k

text: k

text: if

text: r

text: then

text: output

text: (X,

text: Y

text: end

text: L

text: R

text: initialize:

text: let

text: s

text: X

text: and

text: s

text: Y

text: while

text: X

text: do

text: run

text: CreateSequence(r

text: T

text: X)

text: to

text: obtain

text: an

text: α-sequence

text: L

text: of

text: length

text: r

text: L

text: L

text: update:

text: X

text: X

text: V

text: (L),

text: L

text: end

text: while

text: Y

text: do

text: run

text: CreateSequence(r

text: T

text: Y

text: to

text: obtain

text: an

text: α-sequence

text: R

text: of

text: length

text: r

text: R

text: R

text: update:

text: Y

text: Y

text: V

text: (R),

text: R

text: end

text: let

text: λ

text: cLr

text: and

text: m

text: whe≥re

text: c−=

text: if

text: exists

text: L

text: R

text: such

text: that

text: d(V

text: (L),

text: V

text: (R))

text: 4λ

text: then

text: output

text: tr(L

text: R,

text: m);

text: else

text: L

text: R

text: output

text: θ

text: θ

text: for

text: arbitrary:

text: θ

text: and

text: θ

text: end

text: end

text: Algorithm

text: Algorithm

text: CreateSequence

text: Let

text: us

text: assume

text: now

text: that

text: a

text: smooth

text: α-sequence

text: of

text: length

text: k

text: is

text: given.

text: The

text: algorithm

text: FindTrans

text: tries

text: to

text: reconstruct

text: a

text: directed

text: path

text: P

text: by

text: looking

text: for

text: its

text: ith

text: vertex

text: in

text: the

text: matching

text: ordering

text: of

text: P

text: in

text: the

text: k

text: k

text: ith

text: element

text: of

text: the

text: α-sequence

text: θ

text: This

text: is

text: conducted

text: backward

text: edge

text: by

text: backward

text: edge

text: in

text: the

text: matching

text: s

text: ordering

text: of

text: P

text: If

text: the

text: backward

text: edge

text: is

text: not

text: found

text: then

text: two

text: linear-size

text: subsets

text: A

text: A

text: of

text: two

text: distinct

text: k

text: u

text: v

text: elements

text: from

text: the

text: original

text: α-sequence

text: such

text: that

text: A

text: is

text: complete

text: to

text: A

text: are

text: detected.

text: Otherwise

text: an

text: u

text: v

text: α-sequence

text: is

text: updated.

text: The

text: update

text: is

text: done

text: in

text: such

text: a

text: way

text: that

text: if

text: in

text: the

text: new

text: α-sequence

text: the

text: other

text: backward

text: edges

text: of

text: the

text: matching

text: ordering

text: of

text: P

text: are

text: found

text: then

text: they

text: can

text: be

text: combined

text: with

text: the

text: edges

text: k

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: that were already found to reconstruct a copy of P . Since a tournament T the algorithm is working k on is P -free, at some point of its execution two subsets A , A mentioned above with d(A , A ) = 1 k u v u |v will be detected. When that happens, the algorithm is run recursively on the tournaments: T X and | T Y and later two transitive subsets found in these two recursive runs are merged. Let us discuss now subprocedure CreateSequence (Algorithm 3) that constructs an α-sequence of a speciﬁed length r (without loss of generality we will assume that r = 2w for some w > 0). As mentioned before, the procedure can be applied for any forbidden pattern, not only P . Its main k ingredient is called MakeDensePair and is responsible for constructing two disjoint linear sets X, Y in the P -free tournament such that the directed density d(X, Y ) is very close to one. k { } ∪ ∪ Input: a set S , ..., S such that S ... S induces a P -free tournament, a p-vertex i1 ip { i}1 ip k tournament H with V (H) = h , .., h and parameter n; i1 ip Output: a pair of disjoint sets (X, Y ); begin let λ = 1 and m = λk n; 32k∈4 k∈2 { } for each v S and j i , ..., i let N(v, S ) be: i1 2 p j a set of outneighbors of v from S if (h , h ) is an edge and: j i1 j a set of inneighbors of v from S otherwise; ∈ { j } | | | | let bad(v) be: an arbitrary j i , ..., i such that N(v, S ) < λ S or 2 p j j 0 if such a j does not exist; ∈ if there exists v S such that bad(v ) = 0 then ←0 i1 | | 0 ∈ { } update: S tr(N(v , S ), λ S ) for j i , ..., i ; S ←j { ∈ {0 j }}j 2 p let S : j i , ..., i ; new j S 2 p\ { } run MakeDensePair( , H h , n); new i1 else { ∈ } ∈ { } let P = v S : bad(v) = j for j i , ..., i ; j i1 | | 2 p let j = arg max ∈{ } P and P t = tr(P , m); {0 }j i2,...,ip j j0 j0 let W , W , ... be a partitioning of S into sets of size m; 1 2 ≥ j0 if d(P , S ) 1 then j0 j0 2 output (P t , W ), where l = arg max d(P t , W ); j0 lmax max l j0 l else output (W , P t ), where l = arg max d(W , P t ); lmax j0 max l l j0 end end end Algorithm 4: Algorithm MakeDensePair The procedure CreateSequence acts as follows. First two linear sets X, Y of the P -free tour- ≥ − (cid:28) k nament and with d(X, Y ) 1 λ for some 0 < λ 1 are found with the use of the procedure MakeDensePair. If r = 2 then (X, Y ) is output and the procedure is ends. Otherwise, in both X and Y the α-sequences of length r are constructed recursively. When the sequence is constructed, 2 it is deleted from X or Y and a new sequence is being constructed in the remaining set. This is repeated as long there are at least half of the vertices left in X or Y . Let X , X , ... denote the sets of 1 2 the vertices of the α-sequences constructed in X and let Y , Y , ... denote the sets of the vertices of the 1 2 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: that

text: were

text: already

text: found

text: to

text: reconstruct

text: a

text: copy

text: of

text: P

text: Since

text: a

text: tournament

text: T

text: the

text: algorithm

text: is

text: working

text: k

text: on

text: is

text: P

text: -free,

text: at

text: some

text: point

text: of

text: its

text: execution

text: two

text: subsets

text: A

text: A

text: mentioned

text: above

text: with

text: d(A

text: A

text: k

text: u

text: v

text: u

text: |v

text: will

text: be

text: detected.

text: When

text: that

text: happens,

text: the

text: algorithm

text: is

text: run

text: recursively

text: on

text: the

text: tournaments:

text: T

text: X

text: and

text: T

text: Y

text: and

text: later

text: two

text: transitive

text: subsets

text: found

text: in

text: these

text: two

text: recursive

text: runs

text: are

text: merged.

text: Let

text: us

text: discuss

text: now

text: subprocedure

text: CreateSequence

text: (Algorithm

text: that

text: constructs

text: an

text: α-sequence

text: of

text: a

text: speciﬁed

text: length

text: r

text: (without

text: loss

text: of

text: generality

text: we

text: will

text: assume

text: that

text: r

text: for

text: some

text: w

text: As

text: mentioned

text: before,

text: the

text: procedure

text: can

text: be

text: applied

text: for

text: any

text: forbidden

text: pattern,

text: not

text: only

text: P

text: Its

text: main

text: k

text: ingredient

text: is

text: called

text: MakeDensePair

text: and

text: is

text: responsible

text: for

text: constructing

text: two

text: disjoint

text: linear

text: sets

text: X,

text: Y

text: in

text: the

text: P

text: -free

text: tournament

text: such

text: that

text: the

text: directed

text: density

text: d(X,

text: Y

text: is

text: very

text: close

text: to

text: one.

text: k

text: Input:

text: a

text: set

text: S

text: S

text: such

text: that

text: S

text: S

text: induces

text: a

text: P

text: -free

text: tournament,

text: a

text: p-vertex

text: ip

text: ip

text: k

text: tournament

text: H

text: with

text: V

text: (H)

text: h

text: h

text: and

text: parameter

text: n;

text: ip

text: Output:

text: a

text: pair

text: of

text: disjoint

text: sets

text: (X,

text: Y

text: begin

text: let

text: λ

text: and

text: m

text: λk

text: n;

text: for

text: each

text: v

text: S

text: and

text: j

text: i

text: i

text: let

text: N(v,

text: S

text: be:

text: p

text: j

text: a

text: set

text: of

text: outneighbors

text: of

text: v

text: from

text: S

text: if

text: (h

text: h

text: is

text: an

text: edge

text: and:

text: j

text: j

text: a

text: set

text: of

text: inneighbors

text: of

text: v

text: from

text: S

text: otherwise;

text: j

text: let

text: bad(v)

text: be:

text: an

text: arbitrary

text: j

text: i

text: i

text: such

text: that

text: N(v,

text: S

text: λ

text: S

text: or

text: p

text: j

text: j

text: if

text: such

text: a

text: j

text: does

text: not

text: exist;

text: if

text: there

text: exists

text: v

text: S

text: such

text: that

text: bad(v

text: then

text: ←0

text: update:

text: S

text: tr(N(v

text: S

text: λ

text: S

text: for

text: j

text: i

text: i

text: S

text: ←j

text: j

text: }}j

text: p

text: let

text: S

text: j

text: i

text: i

text: new

text: j

text: S

text: p\

text: run

text: MakeDensePair(

text: H

text: h

text: n);

text: new

text: else

text: let

text: P

text: v

text: S

text: bad(v)

text: j

text: for

text: j

text: i

text: i

text: j

text: p

text: let

text: j

text: arg

text: max

text: P

text: and

text: P

text: t

text: tr(P

text: m);

text: }j

text: j

text: let

text: W

text: W

text: be

text: a

text: partitioning

text: of

text: S

text: into

text: sets

text: of

text: size

text: m;

text: if

text: d(P

text: S

text: then

text: output

text: (P

text: t

text: W

text: where

text: l

text: arg

text: max

text: d(P

text: t

text: W

text: lmax

text: max

text: l

text: l

text: else

text: output

text: (W

text: P

text: t

text: where

text: l

text: arg

text: max

text: d(W

text: P

text: t

text: lmax

text: max

text: l

text: l

text: end

text: end

text: end

text: Algorithm

text: Algorithm

text: MakeDensePair

text: The

text: procedure

text: CreateSequence

text: acts

text: as

text: follows.

text: First

text: two

text: linear

text: sets

text: X,

text: Y

text: of

text: the

text: P

text: -free

text: tour-

text: k

text: nament

text: and

text: with

text: d(X,

text: Y

text: λ

text: for

text: some

text: λ

text: are

text: found

text: with

text: the

text: use

text: of

text: the

text: procedure

text: MakeDensePair.

text: If

text: r

text: then

text: (X,

text: Y

text: is

text: output

text: and

text: the

text: procedure

text: is

text: ends.

text: Otherwise,

text: in

text: both

text: X

text: and

text: Y

text: the

text: α-sequences

text: of

text: length

text: r

text: are

text: constructed

text: recursively.

text: When

text: the

text: sequence

text: is

text: constructed,

text: it

text: is

text: deleted

text: from

text: X

text: or

text: Y

text: and

text: a

text: new

text: sequence

text: is

text: being

text: constructed

text: in

text: the

text: remaining

text: set.

text: This

text: is

text: repeated

text: as

text: long

text: there

text: are

text: at

text: least

text: half

text: of

text: the

text: vertices

text: left

text: in

text: X

text: or

text: Y

text: Let

text: X

text: X

text: denote

text: the

text: sets

text: of

text: the

text: vertices

text: of

text: the

text: α-sequences

text: constructed

text: in

text: X

text: and

text: let

text: Y

text: Y

text: denote

text: the

text: sets

text: of

text: the

text: vertices

text: of

text: the

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: [53] K. Zu, R. C. de Lamare and M. Haart, “Generalized design of low-complexity block diagonalization type precoding algorithms for multiuser MIMO systems”, IEEE Trans. Communications, 2013. [54] R. C. de Lamare and A. Alcaim, ”Strategies to improve the performance of very low bit rate speech coders and application to a 1.2 kb/s codec” IEE Proceedings- Vision, image and signal processing, vol. 152, no. 1, February, 2005. [55] Y. Cai, R. C. de Lamare, and R. Fa, “Switched Interleaving Techniques with Limited Feedback for Interference Mitigation in DS-CDMA Sys- tems,” IEEE Transactions on Communications, vol.59, no.7, pp.1946- 1956, July 2011. [56] Y. Cai, R. C. de Lamare, D. Le Ruyet, “Transmit Processing Techniques Based on Switched Interleaving and Limited Feedback for Interference Mitigation in Multiantenna MC-CDMA Systems,” IEEE Transactions on Vehicular Technology, vol.60, no.4, pp.1559-1570, May 2011. [57] C.B. Peel, B.M. Hochwald, and A.L. Swindlehurst, “A vector- perturbation technique for near capacity multiantenna multiuser communication- part I: channel inversion and regularization,” IEEE Trans. Commun., vol. 53, no. 1, pp. 195-202, Jan. 2005. [58] K. Zu, R. C. de Lamare, and M. Haardt, “Multi-Branch Tomlinson- Harashima Precoding Design for MU-MIMO Systems: Theory and Algorithms,” IEEE Transactions on Communications, vol.62, no.3, pp.939-951, March 2014. [59] L. Zhang, Y. Cai, R. C. de Lamare, M. Zhao, “Robust Multi- branch Tomlinson-Harashima Precoding Design in Amplify-and- Forward MIMO Relay Systems,” IEEE Trasactions on Communica- tions, vol.62, no.10, pp.3476-3490, Oct. 2014 [60] T. Peng, R. C. de Lamare, A. Schmeink, “Adaptive Power Allocation Strategies for DSTC in Cooperative MIMO Networks”, IET Commun., 2013. [61] B. Sirkeci-Mergen, A. Scaglione, “Randomized Space-Time Coding for Distributed Cooperative Communication”, IEEE Transactions on Signal Processing, vol. 55, no. 10, Oct. 2007. [62] T. Peng, R. C. de Lamare and A. Schmeink, “Adaptive Distributed Space-Time Coding Based on Adjustable Code Matrices for Cooper- ative MIMO Relaying Systems”, IEEE Transactions on Communica- tions, vol. 61, no. 7, July 2013. [63] B. Hassibi and B. Hochwald, “High-rate codes that are linear in space and time”, IEEE Trans. on Information Theory, vol. 48, Issue 7, pp. 1804-1824, July 2002. [64] J. Liu, R. C. de Lamare, “Low-Latency Reweighted Belief Propagation Decoding for LDPC Codes,” IEEE Communications Letters, vol. 16, no. 10, pp. 1660-1663, October 2012.

text: K.

text: Zu,

text: R.

text: C.

text: de

text: Lamare

text: and

text: M.

text: Haart,

text: “Generalized

text: design

text: of

text: low-complexity

text: block

text: diagonalization

text: type

text: precoding

text: algorithms

text: for

text: multiuser

text: MIMO

text: systems”,

text: IEEE

text: Trans.

text: Communications,

text: R.

text: C.

text: de

text: Lamare

text: and

text: A.

text: Alcaim,

text: ”Strategies

text: to

text: improve

text: the

text: performance

text: of

text: very

text: low

text: bit

text: rate

text: speech

text: coders

text: and

text: application

text: to

text: a

text: kb/s

text: codec”

text: IEE

text: Proceedings-

text: Vision,

text: image

text: and

text: signal

text: processing,

text: vol.

text: no.

text: February,

text: Y.

text: Cai,

text: R.

text: C.

text: de

text: Lamare,

text: and

text: R.

text: Fa,

text: “Switched

text: Interleaving

text: Techniques

text: with

text: Limited

text: Feedback

text: for

text: Interference

text: Mitigation

text: in

text: DS-CDMA

text: Sys-

text: tems,”

text: IEEE

text: Transactions

text: on

text: Communications,

text: July

text: Y.

text: Cai,

text: R.

text: C.

text: de

text: Lamare,

text: D.

text: Le

text: Ruyet,

text: “Transmit

text: Processing

text: Techniques

text: Based

text: on

text: Switched

text: Interleaving

text: and

text: Limited

text: Feedback

text: for

text: Interference

text: Mitigation

text: in

text: Multiantenna

text: MC-CDMA

text: Systems,”

text: IEEE

text: Transactions

text: on

text: Vehicular

text: Technology,

text: May

text: C.B.

text: Peel,

text: B.M.

text: Hochwald,

text: and

text: A.L.

text: Swindlehurst,

text: “A

text: vector-

text: perturbation

text: technique

text: for

text: near

text: capacity

text: multiantenna

text: multiuser

text: communication-

text: part

text: I:

text: channel

text: inversion

text: and

text: regularization,”

text: IEEE

text: Trans.

text: Commun.,

text: vol.

text: no.

text: pp.

text: Jan.

text: K.

text: Zu,

text: R.

text: C.

text: de

text: Lamare,

text: and

text: M.

text: Haardt,

text: “Multi-Branch

text: Tomlinson-

text: Harashima

text: Precoding

text: Design

text: for

text: MU-MIMO

text: Systems:

text: Theory

text: and

text: Algorithms,”

text: IEEE

text: Transactions

text: on

text: Communications,

text: March

text: L.

text: Zhang,

text: Y.

text: Cai,

text: R.

text: C.

text: de

text: Lamare,

text: M.

text: Zhao,

text: “Robust

text: Multi-

text: branch

text: Tomlinson-Harashima

text: Precoding

text: Design

text: in

text: Amplify-and-

text: Forward

text: MIMO

text: Relay

text: Systems,”

text: IEEE

text: Trasactions

text: on

text: Communica-

text: tions,

text: Oct.

text: T.

text: Peng,

text: R.

text: C.

text: de

text: Lamare,

text: A.

text: Schmeink,

text: “Adaptive

text: Power

text: Allocation

text: Strategies

text: for

text: DSTC

text: in

text: Cooperative

text: MIMO

text: Networks”,

text: IET

text: Commun.,

text: B.

text: Sirkeci-Mergen,

text: A.

text: Scaglione,

text: “Randomized

text: Space-Time

text: Coding

text: for

text: Distributed

text: Cooperative

text: Communication”,

text: IEEE

text: Transactions

text: on

text: Signal

text: Processing,

text: vol.

text: no.

text: Oct.

text: T.

text: Peng,

text: R.

text: C.

text: de

text: Lamare

text: and

text: A.

text: Schmeink,

text: “Adaptive

text: Distributed

text: Space-Time

text: Coding

text: Based

text: on

text: Adjustable

text: Code

text: Matrices

text: for

text: Cooper-

text: ative

text: MIMO

text: Relaying

text: Systems”,

text: IEEE

text: Transactions

text: on

text: Communica-

text: tions,

text: vol.

text: no.

text: July

text: B.

text: Hassibi

text: and

text: B.

text: Hochwald,

text: “High-rate

text: codes

text: that

text: are

text: linear

text: in

text: space

text: and

text: time”,

text: IEEE

text: Trans.

text: on

text: Information

text: Theory,

text: vol.

text: Issue

text: pp.

text: July

text: J.

text: Liu,

text: R.

text: C.

text: de

text: Lamare,

text: “Low-Latency

text: Reweighted

text: Belief

text: Propagation

text: Decoding

text: for

text: LDPC

text: Codes,”

text: IEEE

text: Communications

text: Letters,

text: vol.

text: no.

text: pp.

text: October


text: entering (3)), with the power s given by (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) s = 0 : z , z , w , c, 0 + 3 (cid:98) s = 1 : z−, w , w− , 1 1 s = 2 : w− .(cid:101) (cid:101) (cid:101) (cid:101) (39) 3 ± ± The contracted 8-generator algebra expressed by z±, z , c, w (k = 1, 3) is a subalgebra of 0 k the full 12-generator symmetry algebra of the ω = 3 decoupled system with generators given by (31) and (37). The id(cid:101)entiﬁcatio(cid:101)n goes(cid:101)as follows z(cid:101) = eS(cid:101)z e−S = e2it((cid:101)∂ + ix∂ + 3iy∂ + ix2 + 2i), + + t x y (cid:101)z = eS(cid:101)(2id − 3ic)e(cid:101)−S = ∂ , 0 2 t (cid:101)z− = eS(cid:101)(12ir− (cid:101))e−S = 12ie−2ity(∂ + x), 1 x w(cid:101) = eS(cid:101)w e−S = e3(cid:101)it∂ , +3 +3 y w(cid:101) = eS(cid:101)(−2iw )e−S(cid:101) = 2ieit(∂ + x), +1 +1 x w(cid:101)− = eS(cid:101)(−4iw− )e−(cid:101)S = 4ie−it(∂ − x), 1 1 x w−(cid:101) = eS(cid:101)(48w(cid:101)− )e−S = −48e−3ity, 3 3 − c = eSce S = 1, (cid:101) (40) − with the similarity transformation given by S = 3it. 2 The contraction algebra corresponds to the two-dimensional Euclidean algebra acting on ⊃ two sets of creation/annihilation operators. We have the e(2)+h(2) algebra, with non-vanishing commutators given by (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) ±(cid:101) [z , z±] = 2iz±, 0 (cid:101) (cid:101) (cid:101) [z , w ] = kw , 0 k k (cid:101) (cid:101) − (cid:101) [z , w− ] = 4iw , + 1 +1 (cid:101) (cid:101) (cid:101) [z−, w ] = 6iw , +3 +1 (cid:101) (cid:101) (cid:101) [z−, w− ] = 2iw− , 1 3 − | | [w| |, w−| |] = (3 2 k )16c. (41) k k 7 Non-hermitian deformed oscillators with real eigenvalues The non-hermitian operator derived from the degree 0 invariant equation (11) is −1 1 3 H (γ) = ∂ 2 + x2 + 3y∂ + iγx∂ + . (42) 0 x y y 2 2 2 It may be written (see (3)) in the form (cid:98) (cid:98) (cid:98) (cid:98) γ2 − H (γ) = (w− w w− w ) + 2. (43) 0 1 +1 3 +3 16 A discrete spectrum can be calculated algebraically from a lowest weight representation. The ground state ψ is deﬁned by 0,0 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: entering

text: with

text: the

text: power

text: s

text: given

text: by

text: s

text: z

text: z

text: w

text: c,

text: s

text: z−,

text: w

text: w−

text: s

text: w−

text: The

text: contracted

text: algebra

text: expressed

text: by

text: z±,

text: z

text: c,

text: w

text: (k

text: is

text: a

text: subalgebra

text: of

text: k

text: the

text: full

text: symmetry

text: algebra

text: of

text: the

text: ω

text: decoupled

text: system

text: with

text: generators

text: given

text: by

text: and

text: The

text: follows

text: e−S

text: ix∂

text: t

text: x

text: y

text: t

text: x),

text: x

text: e−S

text: y

text: x),

text: x

text: x),

text: x

text: )e−S

text: c

text: eSce

text: S

text: with

text: the

text: similarity

text: transformation

text: given

text: by

text: S

text: The

text: contraction

text: algebra

text: corresponds

text: to

text: the

text: two-dimensional

text: Euclidean

text: algebra

text: acting

text: on

text: two

text: sets

text: of

text: creation/annihilation

text: operators.

text: We

text: have

text: the

text: algebra,

text: with

text: non-vanishing

text: commutators

text: given

text: by

text: [z

text: z±]

text: [z

text: w

text: kw

text: k

text: k

text: [z

text: w−

text: [z−,

text: w

text: [z−,

text: w−

text: [w|

text: w−|

text: k

text: k

text: k

text: Non-hermitian

text: deformed

text: oscillators

text: with

text: real

text: eigenvalues

text: The

text: non-hermitian

text: operator

text: derived

text: from

text: the

text: degree

text: invariant

text: equation

text: is

text: −1

text: H

text: (γ)

text: iγx∂

text: x

text: y

text: y

text: It

text: may

text: be

text: written

text: (see

text: in

text: the

text: form

text: γ2

text: H

text: (γ)

text: (w−

text: w

text: w−

text: w

text: A

text: discrete

text: spectrum

text: can

text: be

text: calculated

text: algebraically

text: from

text: a

text: lowest

text: weight

text: representation.

text: The

text: ground

text: state

text: ψ

text: is

text: deﬁned

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:99) − For generic half-integer (cid:96), the number of non-vanishing deformation parameters γ is (cid:96) 1. j 2 For (cid:96) = 5 the arithmetic progression of ω frequencies entering eq. (65) and producing the cga 2 ± ± ±i (cid:96)=5 2 symmetry algebra is given by 1, 3, 5. The 1, 3, 5 sequence corresponds to the bounded − spectrum, while the 1, 3, 5 sequence corresponds to a special case of three Pais-Uhlenbeck oscillators. N The extension of this construction to the (cid:96) = 1 + centrally extended Conformal Galilei 2 0 Algebras with d > 1 (see [3]) is immediate. The invariant PDEs with discrete spectrum corre- spond to non-hermitian operators whose spectrum is real and given by d copies of the energy modes created in the d = 1 case. In the class of systems here investigated, the oscillators turn out to have all diﬀerent fre- quencies. Equal frequency oscillators are obtained from the diﬀerent class of second-order PDEs invariant under the d = 2 centrally extended Conformal Galilei Algebras with integer (cid:96), see [31]. Acknowledgements We thank T. Yumibayashi for helpful discussions on Pais-Uhlenbeck oscillators. Z.K. and F.T. are grateful to the Osaka Prefecture University, where this work was elaborated, for hos- pitality. F.T. received support from CNPq (PQ Grant No. 306333/2013-9). N. A. is supported by the grants-in-aid from JSPS (Contract No. 26400209). References [1] N. Aizawa, Y. Kimura, and J. Segar, J. Phys. A: Math. Theor. 46, 405204 (2013). [2] N. Aizawa, Z. Kuznetsova, and F. Toppan, J. Math. Phys. 56, 031701 (2015). [3] J. Negro, M. Del Olmo, and A. Rodrıguez-Marco, J. Math. Phys. 38, 3786 (1997). [4] A. Smilga, SIGMA 5, 017 (2009). [5] A. Pais and G. Uhlenbeck, Phys. Rev. 79, 145 (1950). [6] S. V. Ketov, G. Michiaki, and T. Yumibayashi, Quantizing with a higher time derivative, in Advances in Quantum Field Theory, edited by S. Ketov, InTech Publishers, p. 49 (2012). [7] M. Ostrogradski˘ı, Mem. Ac. St. Petersbourg VI, 385 (1850). [8] A. Galajinsky and I. Masterov, Phys. Lett. B 723, 190 (2013). [9] K. Andrzejewski, A. Galajinsky, J. Gonera, and I. Masterov, Nucl. Phys. B 885, 150 (2014). [10] K. Andrzejewski, Phys. Lett. B 738, 405 (2014). [11] K. Andrzejewski, Nucl. Phys. B 889, 333 (2014). [12] A. Galajinsky and I. Masterov, Nucl. Phys. B 896, 244 (2015). [13] J. Gomis and K. Kamimura, Physical Review D 85, 045023 (2012). [14] U. Niederer, Helv. Phys. Acta 45, 802 (1972). [15] F. Toppan, J. Phys. Conf. Ser. 597, 012071 (2015). [16] U. Niederer, Helv. Phys. Acta 46, 191 (1973). 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: For

text: generic

text: half-integer

text: the

text: number

text: of

text: non-vanishing

text: deformation

text: parameters

text: γ

text: is

text: j

text: For

text: the

text: arithmetic

text: progression

text: of

text: ω

text: frequencies

text: entering

text: eq.

text: and

text: producing

text: the

text: cga

text: ±i

text: symmetry

text: algebra

text: is

text: given

text: by

text: The

text: sequence

text: corresponds

text: to

text: the

text: bounded

text: spectrum,

text: while

text: the

text: sequence

text: corresponds

text: to

text: a

text: special

text: case

text: of

text: three

text: Pais-Uhlenbeck

text: oscillators.

text: N

text: The

text: extension

text: of

text: this

text: construction

text: to

text: the

text: centrally

text: extended

text: Conformal

text: Galilei

text: Algebras

text: with

text: d

text: (see

text: is

text: immediate.

text: The

text: invariant

text: PDEs

text: with

text: discrete

text: spectrum

text: corre-

text: spond

text: to

text: non-hermitian

text: operators

text: whose

text: spectrum

text: is

text: real

text: and

text: given

text: by

text: d

text: copies

text: of

text: the

text: energy

text: modes

text: created

text: in

text: the

text: d

text: case.

text: In

text: the

text: class

text: of

text: systems

text: here

text: investigated,

text: the

text: oscillators

text: turn

text: out

text: to

text: have

text: all

text: diﬀerent

text: fre-

text: quencies.

text: Equal

text: frequency

text: oscillators

text: are

text: obtained

text: from

text: the

text: diﬀerent

text: class

text: of

text: second-order

text: PDEs

text: invariant

text: under

text: the

text: d

text: centrally

text: extended

text: Conformal

text: Galilei

text: Algebras

text: with

text: integer

text: see

text: Acknowledgements

text: We

text: thank

text: T.

text: Yumibayashi

text: for

text: helpful

text: discussions

text: on

text: Pais-Uhlenbeck

text: oscillators.

text: Z.K.

text: and

text: F.T.

text: are

text: grateful

text: to

text: the

text: Osaka

text: Prefecture

text: University,

text: where

text: this

text: work

text: was

text: elaborated,

text: for

text: hos-

text: pitality.

text: F.T.

text: received

text: support

text: from

text: CNPq

text: (PQ

text: Grant

text: No.

text: N.

text: A.

text: is

text: supported

text: by

text: the

text: grants-in-aid

text: from

text: JSPS

text: (Contract

text: No.

text: References

text: N.

text: Aizawa,

text: Y.

text: Kimura,

text: and

text: J.

text: Segar,

text: J.

text: Phys.

text: A:

text: Math.

text: Theor.

text: N.

text: Aizawa,

text: Z.

text: Kuznetsova,

text: and

text: F.

text: Toppan,

text: J.

text: Math.

text: Phys.

text: J.

text: Negro,

text: M.

text: Del

text: Olmo,

text: and

text: A.

text: Rodrıguez-Marco,

text: J.

text: Math.

text: Phys.

text: A.

text: Smilga,

text: SIGMA

text: A.

text: Pais

text: and

text: G.

text: Uhlenbeck,

text: Phys.

text: Rev.

text: S.

text: V.

text: Ketov,

text: G.

text: Michiaki,

text: and

text: T.

text: Yumibayashi,

text: Quantizing

text: with

text: a

text: higher

text: time

text: derivative,

text: in

text: Advances

text: in

text: Quantum

text: Field

text: Theory,

text: edited

text: by

text: S.

text: Ketov,

text: InTech

text: Publishers,

text: p.

text: M.

text: Ostrogradski˘ı,

text: Mem.

text: Ac.

text: St.

text: Petersbourg

text: VI,

text: A.

text: Galajinsky

text: and

text: I.

text: Masterov,

text: Phys.

text: Lett.

text: B

text: K.

text: Andrzejewski,

text: A.

text: Galajinsky,

text: J.

text: Gonera,

text: and

text: I.

text: Masterov,

text: Nucl.

text: Phys.

text: B

text: K.

text: Andrzejewski,

text: Phys.

text: Lett.

text: B

text: K.

text: Andrzejewski,

text: Nucl.

text: Phys.

text: B

text: A.

text: Galajinsky

text: and

text: I.

text: Masterov,

text: Nucl.

text: Phys.

text: B

text: J.

text: Gomis

text: and

text: K.

text: Kamimura,

text: Physical

text: Review

text: D

text: U.

text: Niederer,

text: Helv.

text: Phys.

text: Acta

text: F.

text: Toppan,

text: J.

text: Phys.

text: Conf.

text: Ser.

text: U.

text: Niederer,

text: Helv.

text: Phys.

text: Acta

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##