On the optimality of semidefinite relaxations for average-case and generalized constraint satisfaction

Boaz Barak, Guy Kindler, David Steurer

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

This work studies several questions about the optimality of semidefinite programming (SDP) for constraint satisfaction problems (CSPs). First we propose the hypothesis that the well known Basic SDP relaxation is actually optimal for random instances of constraint satisfaction problems for every predicate. This unifies several conjectures proposed in the past, and suggests a unifying principle for the average-case complexity of CSPs. We provide several types of indirect evidence for the truth of this hypothesis, and also show that it (and its variants) imply several conjectures in hardness of approximation including polynomial factor hardness for the densest k subgraph problem and hard instances for the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell (1993). Second, we observe that for every predicate P, the basic SDP relaxation achieves the same approximation guarantee for the CSP for P and for a more general problem (involving not just Boolean but constrained vector assignments), which we call the Generalized CSP for P. Raghavendra (2008) showed that it is UGC-hard to approximate the CSP for P better than this guarantee. We show that it is NP-hard to approximate the Generalized CSP for P better than this guarantee.

Original languageAmerican English
Title of host publicationITCS 2013 - Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science
Pages197-213
Number of pages17
DOIs
StatePublished - 2013
Event2013 4th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2013 - Berkeley, CA, United States
Duration: 9 Jan 201312 Jan 2013

Publication series

NameITCS 2013 - Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science

Conference

Conference2013 4th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2013
Country/TerritoryUnited States
CityBerkeley, CA
Period9/01/1312/01/13

Keywords

  • UGC
  • complexity
  • hardness of approximation
  • semi-definite program
  • unique games conjecture

All Science Journal Classification (ASJC) codes

  • Management of Technology and Innovation
  • Computer Science (miscellaneous)

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