Dependent Randomized Rounding for Matroid Polytopes and Applications

Motivated by several applications, we consider the problem of randomly rounding a fractional solutionin a matroid (base) polytope to an integral one. We consider the pipage rounding technique [5, 6, 36] andalso present a new technique, randomized swap rounding. Our main technical results are concentrationbounds for functions of random variables arising from these rounding techniques. We prove Chernoff-type concentration bounds for linear functions of random variables arising from both techniques, and alsoa lower-tail exponential bound for monotone submodular functions of variables arising from randomizedswap rounding.The following are examples of our applications.• We give a (1−1/e−ε)-approximation algorithm for the problem of maximizing a monotone submod-ular function subject to 1 matroid and k linear constraints, for any constant k ≥ 1 and ε > 0. We alsogive the same result for a super-constant number k of ”loose” linear constraints, where the right-handside dominates the matrix entries by an Ω(ε−2 log k) factor.• We present a result on minimax packing problems that involve a matroid base constraint. We givean O(logm/ log logm)-approximation for the general problem min{λ : ∃x ∈ {0, 1} , x ∈ B(M),Ax ≤ λb} where m is the number of packing constraints. Examples include the low-congestionmulti-path routing problem [34] and spanning-tree problems with capacity constraints on cuts [4, 16].• We generalize the continuous greedy algorithm [35, 6] to problems involving multiple submodularfunctions, and use it to find a (1 − 1/e − ε)-approximate pareto set for the problem of maximizinga constant number of monotone submodular functions subject to a matroid constraint. An example isthe Submodular Welfare Problem where we are looking for an approximate pareto set with respect toindividual players’ utilities. ∗Dept. of Computer Science, Univ. of Illinois, Urbana, IL 61801. Partially supported by NSF grant CCF-0728782. E-mail:[email protected]†IBM Almaden Research Center, San Jose, CA 95120. E-mail: [email protected]‡Institute for Operations Research, ETH Zurich. E-mail: [email protected]