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On the usefulness of predicates

Published: 29 May 2013 Publication History

Abstract

Motivated by the pervasiveness of strong inapproximability 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 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. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max-P, it is hard to find an assignment such that the induced distribution on k-bit strings defined by the instance is not essentially uniform.
Under the unique games conjecture, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise 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.

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    Published In

    cover image ACM Transactions on Computation Theory
    ACM Transactions on Computation Theory  Volume 5, Issue 1
    May 2013
    58 pages
    ISSN:1942-3454
    EISSN:1942-3462
    DOI:10.1145/2462896
    Issue’s Table of Contents
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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    Publication History

    Published: 29 May 2013
    Accepted: 01 March 2013
    Revised: 01 January 2013
    Received: 01 April 2012
    Published in TOCT Volume 5, Issue 1

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    Author Tags

    1. Constraint satisfaction problem
    2. approximation resistance
    3. unique games conjecture
    4. usefulness

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    • (2021)The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite DomainsACM Transactions on Algorithms10.1145/345804117:3(1-23)Online publication date: 15-Jul-2021
    • (2021)Algebraic Approach to Promise Constraint SatisfactionJournal of the ACM10.1145/345760668:4(1-66)Online publication date: 14-Jul-2021
    • (2019)On the approximation resistance of balanced linear threshold functionsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316374(430-441)Online publication date: 23-Jun-2019
    • (2019)Robust Algorithms with Polynomial Loss for Near-Unanimity CSPsSIAM Journal on Computing10.1137/18M116393248:6(1763-1795)Online publication date: 26-Nov-2019
    • (2017)From weak to strong LP gaps for all CSPsProceedings of the 32nd Computational Complexity Conference10.5555/3135595.3135606(1-27)Online publication date: 9-Jul-2017
    • (2017)Robust algorithms with polynomial loss for near-unanimity CSPsProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3039686.3039708(340-357)Online publication date: 16-Jan-2017
    • (2017)$(2+\varepsilon)$-Sat Is NP-hardSIAM Journal on Computing10.1137/15M100650746:5(1554-1573)Online publication date: Jan-2017
    • (2016)Approximation Resistance from Pairwise-Independent SubgroupsJournal of the ACM10.1145/287305463:3(1-32)Online publication date: 12-Aug-2016
    • (2015)How to Refute a Random CSPProceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2015.48(689-708)Online publication date: 17-Oct-2015
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