Matroid Intersections, Polymatroid Inequalities, and Related Problems

Abstract

Given m matroids M 1,..., M m on the common ground set V, it is shown that all maximal subsets of V, independent in the m matroids, can be generated in quasi-polynomial time. More generally, given a system of polymatroid inequalities f 1(X) ≥ t 1,..., f m(X) ≥ t m with quasi-polynomially bounded right hand sides t 1,..., tm, all minimal feasible solutions X⊆V to the system can be generated in incremental quasi-polynomial time. Our proof of these results is based on a combinatorial inequality for polymatroid functions which may be of independent interest. Precisely, for a polymatroid function f and an integer threshold t ≥ 1, let α = α(f,t) denote the number of maximal sets X ⊆ V satisfying f(X) < t, let β = β(f t) be the number of minimal sets X ⊆ V for which f(X) ≥ t, and let n = |V|. We show that α ≤ maxn,β(logt)/c, where c = c(n,β) is the unique positive root of the equation 2c(n c/logβ - 1) = 1. In particular, our bound implies that α ≤ (nβ)logt. We also give examples of polymatroid functions with arbitrarily large t n,α and β for which α = β(1-01))log t/c.KeywordsBinary TreeProper MappingLinear InequalityMinimal SolutionDisjunctive Normal FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.