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Multi-budgeted Matchings and Matroid Intersection via Dependent Rounding

By et al. Chandra Chekuri


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]

Year: 2011
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