An inequality for polymatroid functions and its applications

An integral-valued set function f : 2 7→ Z is called polymatroid if it is submodular, non-decreasing, and f(∅) = 0. Given 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 if β ≥ 2 then α ≤ β , where c = c(n, β) is the unique positive root of the equation 1 = 2(n log β −1). In particular, our bound implies that α ≤ (nβ) t for all β ≥ 1. We also give examples of polymatroid functions with arbitrarily large t, n, α and β for which α ≥ β log . More generally, given a polymatroid function f : 2 7→ Z and an integral threshold t ≥ 1, consider an arbitrary hypergraph H such that |H| ≥ 2 and f(H) ≥ t for all H ∈ H. Let S be the family of all maximal independent sets X of H for which f(X) < t. Then |S| ≤ |H| . As an application, we show that 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 to this system can be generated in incremental quasi-polynomial time. In contrast to this result, the generation of all maximal infeasible sets is an NP-hard problem for many polymatroid inequalities of small range.