The Moser-Tardos Framework with Partial Resampling

TL;DR

This paper extends the Moser-Tardos resampling algorithm to partial resampling, enabling more efficient solutions for problems involving sums of random variables, with applications in scheduling, graph transversals, and packet routing.

Contribution

It introduces partial resampling in the Moser-Tardos framework, improving algorithmic results for various combinatorial problems.

Findings

Settled a conjecture on graph transversals asymptotically.

Achieved improved approximation ratios for packet routing.

Enhanced algorithmic techniques for problems involving sums of random variables.

Abstract

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006) on graph transversals asymptotically, and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, & Rao (1994).