TITLE: Applications of linear programming duality in combinatorics
QUESTION [19 upvotes]: So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's theorem and Menger's theorem. It's neat that these theorems, all equivalent to one another, can be seen to follow from a single framework.
I was wondering if there are other such interesting applications of LP duality (or more general kinds of duality) in combinatorics, perhaps of a different flavor? Is there a good reference for these topics? The more surprising the applications, the better.

REPLY [5 votes]: Here is a fairly recent example. Davies, Jenssen, Perkins and Roberts use linear programming duality to obtain a tight upper bound on the number of independent sets and matchings in a $d$-regular graph on $n$ vertices. 
In fact, more generally, their proof actually gives a bound on the so called "independence polynomial" and the "matching polynomial" of $d$-regular graphs on $n$ vertices. Basically, (I think) their proof uses linear programming duality to provide bounds on the logarithmic derivative of this polynomial and translates this to give bounds on the polynomial itself. 
In the case of independent sets, their result strengthens a result of Zhao (which, itself, extended a result of Galvin and Tetali, which strengthened a result of Kahn). For matchings, their result proves the asymptotic Upper Matching Conjecture of Friedland, Krop, Lundow and Markström.
Yufei Zhao has written a blog post and a survey paper about this topic.