An inequality for polymatroid functions and its applications

An integral-valued set function f:2v ↦ Z is called polymatroid if it is submodular, nondecreasing, and f(φ) = 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(log t)/c, where c = c(n,β) is the unique positive root of the equation 1 = 2c(nc/log β - 1). In particular, our bound implies that α ≤ (nβ)log t for all β ≥ 1. We also give examples of polymatroid functions with arbitrarily large t, n, α and β for which α ≥ β(0.551 log t)/c. More generally, given a polymatroid function f : 2v ↦ 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 f' be the family of all maximal independent sets X of H' for which f(X) < t. Then |f'| ≤ |H'|(log t)/c(n,|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.