Invariants of Binomial Edge Ideals via Linear Programs

We associate to every graph a linear program for packings of vertex disjoint paths. We show that the optimal primal and dual values of the corresponding integer program are the binomial grade and height of the binomial edge ideal of the graph. We deduce from this a new combinatorial characterization of graphs of K\"onig type and use it to show that all trees are of K\"onig type. The log canonical threshold and the F-threshold are important invariants associated to the singularities of a variety in characteristic $0$ and characteristic $p$. We show that the optimal value of the linear program (computed over the rationals) agrees with both the F-threshold and the log canonical threshold of the binomial edge ideal if the graph is a block graph or of K\"onig type. We conjecture that this linear program computes the log canonical threshold of the binomial edge ideal of any graph. Our results resemble theorems on monomial ideals arising from hypergraphs due to Howald and others.