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_29: We give a worst-case analysis for two greedy heuristics for the integer programming problem minimize cx , Ax (ge) b , 0 (le) x (le) u , x integer, where the entries in A, b , and c are all nonnegative. The first heuristic is for the case where the entries in A and b are integral, the second only assumes the rows are scaled so that the smallest nonzero entry is at least 1. In both cases we compare the ratio of the value of the greedy solution to that of the integer optimal. The error bound grows logarithmically in the maximum column sum of A for both heuristics.
  @cite_7: Worst-case bounds are given on the performance of the greedy heuristic for a continuous version of the set covering problem. This generalizes results of Chvatal, Johnson and Lovasz for the 0-1 covering problem. The results for the greedy heuristic and for other heuristics are obtained by treating the covering problem as a limiting case of a generalized location problem for which worst-case results are known. An alternative approach involving dual greedy heuristics leads also to worst-case bounds for continuous packing problems.
summary: A key corollary of our results is that for families of instances of CIPs, we get a good ( @math or @math ) integrality gap if @math grows at least as fast as @math . Bounds on the result of a greedy algorithm for CIPs relative to the optimal solution, are known @cite_29 @cite_7 . Our bound improves that of @cite_29 and is incomparable with @cite_7 ; for any given @math , @math , and the unit vector @math , our bound improves on @cite_7 if @math is more than a certain threshold. As it stands, randomized rounding produces such improved solutions for several CIPs only with a very low, sometimes exponentially small, probability. Thus, it does not imply a randomized algorithm, often. To this end, we generalize Raghavan's method of pessimistic estimators to derive an algorithmic (polynomial-time) version of our results for CIPs, in .