Matroid Intersections, Polymatroid Inequalities, and Related Problems (original) (raw)

2002, Lecture Notes in Computer Science

Given m matroids M1, . . . , Mm 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 f1(X) ≥ t1, . . . , fm(X) ≥ tm with quasi-polynomially bounded right hand sides t1, . . . , 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 ≤ max{n, β (log t)/c }, where c = c(n, β) is the unique positive root of the equation 2 c (n c/ log β − 1) = 1. In particular, our bound implies that α ≤ (nβ) log t . We also give examples of polymatroid functions with arbitrarily large t, n, α and β for which α = β (1−o(1)) log t/c .