PCPs via the low-degree long code and hardness for constrained hypergraph coloring∗

We develop new techniques to incorporate the recently proposed “short code” (a low-degree version of the long code) into the construction and analysis of PCPs in the classical “LABEL-COVER + Fourier Analysis” framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs and certain coloring-type problems. In particular, we show a hardness for a variant of hypergraph coloring (with hyperedges of size 6), with a gap between 2 and exp(2 √ log log ) number of colors where N is the number of vertices. This is the first hardness result to go beyond the O(logN) barrier for a coloring-type problem. Our hardness bound is a doubly exponential improvement over the previously known O(log logN)-coloring hardness for 2-colorable hypergraphs, and an exponential improvement over the (logN)-coloring hardness for O(1)-colorable hypergraphs. Stated in terms of “covering complexity,” we show that for 6-ary Boolean CSPs, it is hard to decide if a given instance is perfectly satisfiable or if it requires more than 2 √ log log N) assignments for covering all of the constraints. While our methods do not yield a result for conventional hypergraph coloring due to some technical reasons, we also prove hardness of (logN)-coloring 2-colorable 8-uniform hypergraphs (this result relies just on the long code). A key algebraic result driving our analysis concerns a very low-soundness error testing method for Reed-Muller codes. We prove that if a function β : F2 → F2 is 2 far in absolute distance from polynomials of degree m − d, then the probability that deg(βg) 6 m − 3d/4 for a random degree d/4 polynomial g is doubly exponentially small in d. ∗An extended abstract of this work was presented at the 54th Annual Symposium on Foundations of Computer Science (FOCS), October 2013 [8]. †Department of Applied Math and Computer Science, The Weizmann Institute of Science, Rehovot, Israel. Email: 〈[email protected]〉. Research supported by US-Israel BSF grant number 2008293 and ERC grant number 239985. ‡Computer Science Department, Carnegie Mellon University, Pittsburgh, USA. Email: 〈[email protected]〉. Research supported in part by US-Israel BSF grant number 2008293 and the US National Science Foundation under Grant No. CCF-1115525. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 122 (2013)