TY - GEN

T1 - On the usefulness of predicates

AU - Austrin, Per

AU - Håstad, Johan

PY - 2012

Y1 - 2012

N2 - Motivated by the pervasiveness of strong in approximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate P being \emph{useful} for a (real-valued) objective Q: given an almost satisfiable Max-P instance, there is an algorithm that beats a random assignment on the corresponding Max-Q instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate P is exactly the same as saying that P is useful for P itself. We say that P is useless if it is not useful for any Q. Under the Unique Games Conjecture, we can give a complete and simple characterization of useless Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pair wise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs.

AB - Motivated by the pervasiveness of strong in approximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate P being \emph{useful} for a (real-valued) objective Q: given an almost satisfiable Max-P instance, there is an algorithm that beats a random assignment on the corresponding Max-Q instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicate P is exactly the same as saying that P is useful for P itself. We say that P is useless if it is not useful for any Q. Under the Unique Games Conjecture, we can give a complete and simple characterization of useless Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pair wise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs.

UR - http://www.scopus.com/inward/record.url?scp=84866503497&partnerID=8YFLogxK

U2 - 10.1109/CCC.2012.18

DO - 10.1109/CCC.2012.18

M3 - Conference article in proceedings

AN - SCOPUS:84866503497

SN - 9780769547084

T3 - IEEE Conference on Computational Complexity

SP - 53

EP - 63

BT - Proceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012

T2 - IEEE Computer Society Technical Committee on Mathematical Foundations of Computing

Y2 - 26 June 2012 through 29 June 2012

ER -