TY - GEN

T1 - Approximation algorithms for label cover and the log-density threshold

AU - Chlamtáč, Eden

AU - Manurangsi, Pasin

AU - Moshkovitz, Dana

AU - Vijayaraghavan, Aravindan

N1 - Funding Information:
Department of Computer Science, Ben-Gurion University. Partially supported by ISF grant 1002/14. 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. Department of Computer Science, UT Austin. This material is based upon work supported by the National Science Foundation under grants number 1218547 and 1648712. Department of Electrical Engineering and Computer Science, Northwestern University. Partially supported by NSF grant CCF-1637585.
Publisher Copyright:
Copyright © by SIAM.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85016199453&partnerID=8YFLogxK

U2 - 10.1137/1.9781611974782.57

DO - 10.1137/1.9781611974782.57

M3 - Conference contribution

AN - SCOPUS:85016199453

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 900

EP - 919

BT - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

A2 - Klein, Philip N.

PB - Association for Computing Machinery

T2 - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

Y2 - 16 January 2017 through 19 January 2017

ER -