Lift-and-Project Methods for Set Cover and Knapsack

摘要

We study the applicability of lift-and-project methods to the Set Cover and Knapsack problems. Inspired by recent work of Kar-lin, 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 Inn. 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 Lovasz 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) In 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 Lovasz 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.