Lift-and-Project Methods for Set Cover and Knapsack

We study the applicability of lift-and-project methods to the Set Cover and Knapsack problems. Inspired by recent work of Karlin et al. (IPCO 2011), who examined this connection for Knapsack, we consider the applicability and limitations of these methods for Set Cover, as well as extend the existing results for Knapsack. For the Set Cover problem, Cygan et al. (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 et al. (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.