The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover
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Moshkovitz_The projection.pdf
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Checksum (MD5)
91332ab28b1c85ff8f3e89750e9cf351
Author(s)
Moshkovitz Aaronson, Dana Hadar
Date Issued
2012
Journal
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Publisher
Springer-Verlag
Citation
Moshkovitz, Dana. “The Projection Games Conjecture and the NP-Hardness of Ln n-Approximating Set-Cover.” Lecture Notes in Computer Science (2012): 276–287.
Version
Author's final manuscript
Abstract
We suggest the research agenda of establishing new hardness of approximation results based on the “projection games conjecture”, i.e., an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games.
We pursue this line of research by establishing a tight NP-hardness result for the Set-Cover problem. Specifically, we show that under the projection games conjecture (in fact, under a quantitative version of the conjecture that is only slightly beyond the reach of current techniques), it is NP-hard to approximate Set-Cover on instances of size N to within (1 − α)ln N for arbitrarily small α > 0. Our reduction establishes a tight trade-off between the approximation accuracy α and the time required for the approximation 2[superscript NΩ(α)], assuming Sat requires exponential time.
The reduction is obtained by modifying Feige’s reduction. The latter only provides a lower bound of 2[superscript NΩ(α/loglogN)] on the time required for (1 − α)ln N-approximating Set-Cover assuming Sat requires exponential time (note that N[superscript 1/loglogN] = N[superscript o(1)]). The modification uses a combinatorial construction of a bipartite graph in which any coloring of the first side that does not use a color for more than a small fraction of the vertices, makes most vertices on the other side have their neighbors all colored in different colors.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/978-3-642-32512-0_24
