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Lift-and-Project Methods for Set Cover and Knapsack

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Abstract

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 \((1-\varepsilon )\ln n\) at level \(\Omega (n)\) 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 (\(\mathrm{LS}_+\)) reduces the integrality gap for Knapsack to \((1+\varepsilon )\) 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 \(\mathrm{LS}_+\)), and extends the work of Karlin et al., who demonstrated such an improvement only for the more powerful SDP hierarchy of Lasserre. Our \(\mathrm{LS}_+\)-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 \(\mathrm{LS}_+\) relaxations after a constant number of rounds.

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Notes

  1. The size of the relaxation is, more accurately, increased by a factor of \({n \atopwithdelims ()t}^{O(1)}\) over the original relaxation, so it is accurate to say the running time is sub-exponential when \(t = o(n)\).

  2. Here, as before, \(\mathsf{OPT}\) denotes the value of the optimal 0–1 solution.

  3. Note that this is crucial for a maximization problem like Knapsack, while for a minimization problem like Set Cover it does not seem helpful (and indeed, by the integrality gap of Alekhnovich et al. [1], we know it does not help).

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Acknowledgements

We would like to thank Mohammad R. Salavatipour for preliminary discussions on sub-exponential time approximation algorithms in general. We would also like to thank Claire Mathieu for insightful past discussions of the Knapsack-related results in [15].

Author information

Authors and Affiliations

  1. Department of Computer Science, Ben Gurion University, Beer-Sheva, Israel

    Eden Chlamtáč

  2. Department of Computing Science, University of Alberta, Edmonton, Canada

    Zachary Friggstad

  3. Department of Mathematics, Ryerson University, Toronto, Canada

    Konstantinos Georgiou

Authors
  1. Eden Chlamtáč
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  2. Zachary Friggstad
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  3. Konstantinos Georgiou
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Corresponding author

Correspondence to Eden Chlamtáč.

Additional information

This paper expands on and improves over a preliminary version that appeared in WADS 2013 [6].

Eden Chlamtáč: Partially supported by ISF Grant 1002/14.

Zachary Friggstad: This research was supported, in part, thanks to funding from the Canada Research Chairs program and an NSERC Discovery grant. Konstantinos Georgiou: Partially supported by NSERC.

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Chlamtáč, E., Friggstad, Z. & Georgiou, K. Lift-and-Project Methods for Set Cover and Knapsack. Algorithmica 80, 3920–3942 (2018). https://doi.org/10.1007/s00453-018-0427-4

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