Multi-budgeted matchings and matroid intersection via dependent rounding

Abstract

Motivated by multi-budgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution X in the (non-bipartite graph) matching and matroid intersection polytopes. We show that for any fixed δ > 0, a given point X can be rounded to a random solution R such that E[1R] = (1 - δ)x and any linear function of x satisfies dimension-free Chernoff-Hoeffding concentration bounds (the bounds depend on δ and the expectation μ). We build on and adapt the swap rounding scheme in our recent work [9] to achieve this result. Our main contribution is a non-trivial martingale based analysis framework to prove the desired concentration bounds. In this paper we describe two applications. We give a randomized PTAS for matroid intersection and matchings with any fixed number of budget constraints. We also give a deterministic PTAS for the case of matchings. The concentration bounds also yield related results when the number of budget constraints is not fixed. As a second application we obtain an algorithm to compute in polynomial time an ε-approximate Pareto-optimal set for the multi-objective variants of these problems, when the number of objectives is a fixed constant. We rely on a result of Papadimitriou and Yannakakis [26].