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Optimal algorithms and inapproximability results for every CSP?

Published: 17 May 2008 Publication History

Abstract

Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming.
Making this connection precise, we show the following result : If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains.
Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP.
Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.

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    cover image ACM Conferences
    STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
    May 2008
    712 pages
    ISBN:9781605580470
    DOI:10.1145/1374376
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    Published: 17 May 2008

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    Author Tags

    1. constraint satisfaction problem
    2. dictatorship tests
    3. rounding schemes
    4. semidefinite programming
    5. unique games conjecture

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    May 17 - 20, 2008
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    • (2024)Algebraic Approach to ApproximationProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662076(1-14)Online publication date: 8-Jul-2024
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