Dependent Randomized Rounding for Matroid Polytopes and Applications

TL;DR

This paper introduces new randomized rounding techniques for matroid polytopes, provides concentration bounds for these methods, and applies them to develop approximation algorithms for submodular maximization and packing problems.

Contribution

It presents a new randomized swap rounding technique, derives Chernoff-type concentration bounds, and applies these to improve approximation algorithms for submodular and packing problems.

Findings

Chernoff-type bounds for linear functions of the rounding variables

A lower-tail exponential bound for monotone submodular functions

Approximation algorithms with (1-1/e-epsilon) guarantees for submodular maximization

Abstract

Motivated by several applications, we consider the problem of randomly rounding a fractional solution in a matroid (base) polytope to an integral one. We consider the pipage rounding technique and also present a new technique, randomized swap rounding. Our main technical results are concentration bounds 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 also a lower-tail exponential bound for monotone submodular functions of variables arising from randomized swap rounding. The following are examples of our applications: (1) We give a (1-1/e-epsilon)-approximation algorithm for the problem of maximizing a monotone submodular function subject to 1 matroid and k linear constraints, for any constant k and epsilon>0. (2) We present a result on minimax packing problems that involve a matroid base constraint. We give an O(log m / log log m)-approximation for the general problem Min {lambda: x \in {0,1}^N, x \in B(M), Ax <= lambda b}, where m is the number of packing constraints. (3) We generalize the continuous greedy algorithm to problems involving multiple submodular functions, and use it to find a (1-1/e-epsilon)-approximate pareto set for the problem of maximizing a constant number of monotone submodular functions subject to a matroid constraint. An example is the Submodular Welfare Problem where we are looking for an approximate pareto set with respect to individual players' utilities.