TY - GEN

T1 - On the optimality of semidefinite relaxations for average-case and generalized constraint satisfaction

AU - Barak, Boaz

AU - Kindler, Guy

AU - Steurer, David

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - UGC

KW - complexity

KW - hardness of approximation

KW - semi-definite program

KW - unique games conjecture

UR - http://www.scopus.com/inward/record.url?scp=84873339000&partnerID=8YFLogxK

U2 - 10.1145/2422436.2422460

DO - 10.1145/2422436.2422460

M3 - Conference contribution

AN - SCOPUS:84873339000

SN - 9781450318594

T3 - ITCS 2013 - Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science

SP - 197

EP - 213

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

T2 - 2013 4th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2013

Y2 - 9 January 2013 through 12 January 2013

ER -