Abstract

We present concentration bounds for linear functions of random variables arising from the pipage rounding procedure on matroid polytopes. As an application, we give a (1 − 1/e − ɛ)-approximation algorithm for the problem of maximizing a monotone submodular function subject to 1 matroid and k linear constraints, for any constant k ≥ 1 and ɛ> 0. This generalizes the result for k linear constraints by Kulik et al. [11]. We also give the same result for a super-constant number k of ”loose ” linear constraints, where the right-hand side dominates the matrix entries by an Ω(ɛ −2 log k) factor. As another application, we present a general result on minimax packing problems that involve a matroid base constraint. An example is the multi-path routing problem with integer demands for pairs of vertices; the goal is to minimize congestion. We give an O(log m / log log m)approximation for the general problem min{λ: ∃x ∈ {0, 1} N, x ∈ B(M), Ax ≤ λb} where m is the number of packing constraints.

Citations