Approximation Algorithms for Label Cover and The Log-Density Threshold

Many known optimal NP-hardness of approximation results are reductions from a problem called LabelCover. The input is a bipartite graph G = (L,R,E) and each edge e = (x, y) ∈ E carries a projection πe 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 N3−2 √ 2 ≈ N−0.17. We then design, for any ε > 0, a polynomial-time approximation algorithm for semirandom Label-Cover whose approximation ratio is N3−2 √ 2+ε. 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∗[email protected]. Department of Computer Science, Ben-Gurion University. Partially supported by ISF grant 1002/14. †[email protected]. Department of Electrical Engineering and Computer Science, UC Berkeley. This material is based upon work supported by the National Science Foundation under grants number CCF 1540685 and CCF 1655215. ‡[email protected]. Department of Computer Science, UT Austin. This material is based upon work supported by the National Science Foundation under grants number 1218547 and 1648712. §[email protected]. Department of Electrical Engineering and Computer Science, Northwestern University. Partially supported by NSF grant CCF-1637585. 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 NΩ(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+ε for any ε > 0.