An extension of the Lovász local lemma, and its applications to integer programming
Aravind Srinivasan · 1996
The Lov'asz Local Lemma (LLL) 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. We consider three classes of NP-hard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL 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 sparse (e.g., VLSI routing using short paths, problems on hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan, to constructivize our packing and covering results. 1