Approximation algorithms for label cover and the log-density threshold

Many known optimal NP-hardness of approximation results are reductions from a problem called Label- Cover. The input is a bipartite graph G = (L,R,E) and each edge e = (x, y) 2 E carries a projection π e that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for Label-Cover is of the form N-c where N = nk, n is the number of vertices, k is the number of labels, and 0 < c < 1 is some constant. Inspired by a framework originally developed for Densest k-Subgraph, we propose a "log density threshold" for the approximability of Label-Cover. Specifically, we suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is N3-2p2 N-0.17. We then design, for any 0, a polynomial-time approximation algorithm for semirandom Label-Cover whose approximation ratio is N3-2p2+. In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary. For worst-case Label-Cover we show a polynomial- time algorithm whose approximation ratio is roughly N-0.233. The previous best efficient approximation ratio was N-0.25. We present some evidence towards an N-c threshold by constructing integrality gaps for N(1) rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the "log density threshold" is N-0.25, and we give a polynomial-time algorithm in the semi-random model whose approximation ratio is N-0.25+ for any > 0.