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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.

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.

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 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.

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.

Here is a fairly recent example of an application. 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 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.

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 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.