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On the usefulness of predicates
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-5379-345X
2012 (English)In: 2012 IEEE 27th Annual Conference On Computational Complexity (CCC), IEEE , 2012, 53-63 p.Conference paper (Refereed)
Abstract [en]

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.

Place, publisher, year, edition, pages
IEEE , 2012. 53-63 p.
, Proceedings of the Annual IEEE Conference on Computational Complexity, ISSN 1093-0159
Keyword [en]
Approximability, Approximate solution, Literals, MAX-CSP, Random assignment, Satisfying assignments, Unique games conjecture, Computational complexity, Approximation algorithms
National Category
Computer Science
URN: urn:nbn:se:kth:diva-105331DOI: 10.1109/CCC.2012.18ISI: 000308976600006ScopusID: 2-s2.0-84866503497ISBN: 978-0-7695-4708-4OAI: diva2:570627
IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, 26 June 2012 through 29 June 2012, Porto

QC 20121120

Available from: 2012-11-20 Created: 2012-11-20 Last updated: 2012-11-20Bibliographically approved

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