A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
An Explicit Equivalent Positive Semidefinite Program for Nonlinear 0-1 Programs
SIAM Journal on Optimization
Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Exponential-time approximation of weighted set cover
Information Processing Letters
Approximation algorithms for combinatorial problems
Journal of Computer and System Sciences
Integrality gaps of linear and semi-definite programming relaxations for Knapsack
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Rounding Semidefinite Programming Hierarchies via Global Correlation
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Approximating CSPs with global cardinality constraints using SDP hierarchies
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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We study the applicability of lift-and-project methods to the Set Cover and Knapsack problems. Inspired by recent work of Karlin, Mathieu, and Nguyen [IPCO 2011], who examined this connection for Knapsack, we consider the applicability and limitations of these methods for Set Cover, as well as extending extending the existing results for Knapsack. For the Set Cover problem, Cygan, Kowalik, and Wykurz [IPL 2009] gave sub-exponential-time approximation algorithms with approximation ratios better than ln n. We present a very simple combinatorial algorithm which has nearly the same time-approximation tradeoff as the algorithm of Cygan et al. We then adapt this to an LP-based algorithm using the LP hierarchy of Lovász and Schrijver. However, our approach involves the trick of "lifting the objective function". We show that this trick is essential, by demonstrating an integrality gap of (1−ε)ln n at level Ω(n) of the stronger LP hierarchy of Sherali and Adams (when the objective function is not lifted). Finally, we show that the SDP hierarchy of Lovász and Schrijver (LS+) reduces the integrality gap for Knapsack to (1+ε) at level O(1). This stands in contrast to Set Cover (where the work of Aleknovich, Arora, and Tourlakis [STOC 2005] rules out any improvement using LS+), and extends the work of Karlin et al., who demonstrated such an improvement only for the more powerful SDP hierarchy of Lasserre. Our LS+ based rounding and analysis are quite different from theirs (in particular, not relying on the decomposition theorem they prove for the Lasserre hierarchy), and to the best of our knowledge represents the first explicit demonstration of such a reduction in the integrality gap of LS+ relaxations after a constant number of rounds.