An Extension of the Lovasz Local Lemma, and its Applications to Integer Programming

TL;DR

This paper extends the Lovasz Local Lemma to better handle conjunctions of events related to variable deviations, enabling improved approximation guarantees for certain NP-hard integer programs via randomized rounding.

Contribution

It introduces a novel extension of the Lovasz Local Lemma applicable to conjunctions of deviation events, and applies it to derive better approximation results for column-sparse integer programs.

Findings

Randomized rounding yields better feasible solutions with non-zero probability.

The extension improves approximation guarantees for minimax and covering integer programs.

Constructive algorithms are developed using a generalized pessimistic estimators method.

Abstract

The Lovasz Local Lemma due to Erdos and Lovasz is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson to derive good approximation algorithms for such problems. We use our extension of the Local Lemma to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension/degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan, to obtain constructive (algorithmic) versions of our results for covering integer programs.