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 − ǫ)-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 ǫ > 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(ǫ −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.
Chekuri et al. REF give strong concentration results for dependent randomized rounding with many applications; one of these applications is a ((e/e − 1) − ǫ)-approximation for monotone maximization with respect to a matroid and any constant number of knapsack constraints.
Randomized pipage rounding for matroid polytopes and applications
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Mathematics Computer Science