On column-restricted and priority covering integer programs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Approximating sparse covering integer programs online
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Approximating the min-max (regret) selecting items problem
Information Processing Letters
Constraint satisfaction, packet routing, and the lovasz local lemma
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Approximation algorithms for throughput maximization in wireless networks with delay constraints
IEEE/ACM Transactions on Networking (TON)
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The Lova´sz local lemma due to Erdo˝s and Lova´sz (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 and 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 nonzero 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. Comput. System Sci., 37 (1988), pp. 130-143), to obtain constructive (algorithmic) versions of our results for covering integer programs.