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 2^c(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.

This research was supported by the National Science Foundation (Grant IIS-0118635), and by the Office of Naval Research (Grant N00014-92-J-1375). The second and third authors are also grateful for the partial support by DIM ACS, the National Science Foundation’s Center for Discrete Mathematics and Theoretical Computer Science.