The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover

Dana Moshkovitz
2015 Theory of Computing  
We establish a tight NP-hardness result for approximating the SET-COVER problem based on a strong PCP theorem. Our work implies that 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 running time exp(N Ω(α) ) assuming SAT requires exponential time. The reduction is obtained by modifying Feige's reduction. The latter provides a lower bound of
more » ... (N Ω(α/ log log N) ) on the time required for (1 − α) ln N-approximating SET-COVER assuming SAT requires exponential time. 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 all their neighbors colored in different colors. In the conference version of this paper, the SET-COVER result was conditioned on a conjecture we call "The Projection Games Conjecture" (PGC), a strengthening of the Sliding A preliminary version of this and Russell to projection games (LABEL-COVER). More precisely, the prerequisite was a quantitative version of this conjecture that was slightly beyond what was known at the time of the paper's writing. Shortly afterward, Dinur and Steurer, based on a result by the author and Raz, proved the quantitative version of the conjecture sufficient for the SET-COVER result. More broadly, in this paper we discuss the Projection Games Conjecture and its applications to hardness of approximation, e. g., to polynomial hardness factors for the CLOSEST-VECTOR problem and to studying the behavior of CSPs around their approximability threshold.
doi:10.4086/toc.2015.v011a007 dblp:journals/toc/Moshkovitz15 fatcat:sg65cikhdre7vdecuqrsmmvjqi