On Sparsification of Stochastic Packing Problems

TL;DR

This paper studies how to efficiently create sparse representations of stochastic packing problems that preserve near-optimal solutions, with applications to matroids, matchings, and submodular objectives.

Contribution

It introduces a generic sparsifier based on contention resolution and develops improved sparsifiers for matroids and matchings, advancing the understanding of sparsification in stochastic packing.

Findings

A generic sparsifier of degree 1/p matches best contention resolution schemes.

Improved sparsifiers for matroids and weighted matching achieve linear degree in 1/p.

Submodular packing problems do not admit near-optimal sparsifiers.

Abstract

Motivated by recent progress on stochastic matching with few queries, we embark on a systematic study of the sparsification of stochastic packing problems (SPP) more generally. Specifically, we consider SPPs where elements are independently active with a probability p, and ask whether one can (non-adaptively) compute a sparse set of elements guaranteed to contain an approximately optimal solution to the realized (active) subproblem. We seek structural and algorithmic results of broad applicability to such problems. Our focus is on computing sparse sets containing on the order of d feasible solutions to the packing problem, where d is linear or at most poly. in 1/p. Crucially, we require d to be independent of the any parameter related to the ``size'' of the packing problem. We refer to d as the degree of the sparsifier, as is consistent with graph theoretic degree in the special case of matching. First, we exhibit a generic sparsifier of degree 1/p based on contention resolution. This sparsifier's approximation ratio matches the best contention resolution scheme (CRS) for any packing problem for additive objectives, and approximately matches the best monotone CRS for submodular objectives. Second, we embark on outperforming this generic sparsifier for matroids, their intersections and weighted matching. These improved sparsifiers feature different algorithmic and analytic approaches, and have degree linear in 1/p. In the case of a single matroid, our sparsifier tends to the optimal solution. For weighted matching, we combine our contention-resolution-based sparsifier with technical approaches of prior work to improve the state of the art ratio from 0.501 to 0.536. Third, we examine packing problems with submodular objectives. We show that even the simplest such problems do not admit sparsifiers approaching optimality.