text: 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.
  @cite_22: We study the relation between a class of 0–1 integer linear programs and their rational relaxations. We give a randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0–1 problem. Our technique can be a of extended to provide bounds on the disparity between the rational and 0–1 optima for a given problem instance.
summary: To see what problems MIPs model, note, from constraints (i) and (iii) of MIPs, that for all @math , any feasible solution will make the set @math have precisely one 1, with all other elements being 0; MIPs thus model many choice'' scenarios. Consider, e.g., global routing in VLSI gate arrays @cite_22 . Given are an undirected graph @math , a function @math , and @math , a set @math of paths in @math , each connecting @math to @math ; we must connect each @math with @math using exactly one path from @math , so that the maximum number of times that any edge in @math is used for, is minimized--an MIP formulation is obvious, with @math being the indicator variable for picking the @math th path in @math . This problem, the vector-selection problem of @cite_22 , and the discrepancy-type problems of , are all modeled by MIPs; many MIP instances, e.g., global routing, are NP-hard.