An extension of the Lovász local lemma, and its applications to integer programming

Abstract

The Lovasz Local Lemma due to Erdős and Lovasz (in Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 11, 1975, pp. 609–627) 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 & Thompson (Combinatorica, 7 (1987), pp. 365–374 ) 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 (J. Computer and System Sciences, 37 (1988), pp. 130–143 ), to obtain constructive (algorithmic) versions of our results for covering integer programs.