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Circumventing d-to-1 for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width at Least Four
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
2012 (English)In: Electronic Colloquium on Computational Complexity (ECCC), ISSN 1433-8092Article in journal (Refereed) Published
Abstract [en]

Håstad established that any predicate P01m  containing parity of width at least three is approximation resistant for almost satisfiable instances. However, in comparison to for example the approximation hardness of Max-3SAT, the result only holds for almost satisfiable instances. This limitation was addressed by O'Donnell, Wu, and Huang who showed the threshold result that if a predicate strictly contains parity of width at least three, then it is approximation resistant also for satisfiable instances, assuming the d-to-1 Conjecture. We extend modern hardness-of-approximation techniques by Mossel et al. to projection games, eliminating dependencies on the degree of projections via Smooth Label Cover, and prove, subject only to = , the same approximation-resistance result for predicates of width four or greater.

Place, publisher, year, edition, pages
2012.
Keyword [en]
Approximation Resistance, correlations, d-to-1 conjecture, Invariance, Perfect Completeness, Smooth Label cover
National Category
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-113213OAI: oai:DiVA.org:kth-113213DiVA: diva2:587576
Note

QC 20130502

Available from: 2013-01-14 Created: 2013-01-14 Last updated: 2014-09-29Bibliographically approved
In thesis
1. Label Cover Reductions for Unconditional Approximation Hardness of Constraint Satisfaction
Open this publication in new window or tab >>Label Cover Reductions for Unconditional Approximation Hardness of Constraint Satisfaction
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Etikettäckningsreduktioner för Obetingad Approximationssvårighet av Vilkorssatisfiering
Abstract [en]

Problem solving is an integral aspect of modern society and includes such tasks as picking the fastest route to work, optimizing a production line, scheduling computer tasks, placing new bus stops, or picking a meal from available ingredients.We study the hardness of solving Constraint Satisfaction Problems (CSPs). In these, one is given a collection of constraints on variables with the task of finding an assignment satisfying the greatest number of constraints. In particular, parity constraints dictate that an odd (alt. even) number of variables are assigned a certain value.Satisfiable collections of parity constraints are easy in the sense that they can be efficiently solved via Gaussian elimination. We prove the threshold phenomenon that when constraints accept even one more assignment besides parities, then it is hard to find approximate solutions which are essentially better than random assignments.We also investigate the uselessness of predicates. Uselessness is a stronger hardness property in the sense that even if one was permitted to choose the acceptance criteria for given constraints, it is NP-hard to find solutions beating random assignments. We provide the first examples of predicates which are useless even when all variables appear unnegated.Finally, in an Ordering CSP (OCSP), one receives a set of items to order and constraints specifying how the items should be ordered relative to one another. For example, in the problem Maximum Betweenness, we have constraints of the form "order x between y and z". Our contribution is to significantly improve the approximation hardness of various OCSPs and provide the first unconditional direct Probabilistically Checkable Proofs for OCSPs.Notably, all results were previously known assuming the Unique Games Conjecture and the d-to-1 Conjecture. Our unconditional analogues of the same theorems involve developments for dealing with various obstacles faced by conventional techniques.

Place, publisher, year, edition, pages
Stockholm: Numerical Analysis and Computer Science (NADA), Stockholm University, 2014. xii, 52 p.
Keyword
Optimization, NP, Approximation, Approximability, Inapproximability, Constraint Satisfaction, CSP, Boolean Analysis, Satisfiability, SAT, Acyclic Subgraph, Betweenness, Unique Games
National Category
Computer Science
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-151402 (URN)
Public defence
2014-10-21, F3, Sing Sing, Lindstedtsvägen 26, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Projects
Approximation of NP-hard optimization problems
Funder
EU, European Research Council, 226203
Note

QC 20140929

Available from: 2014-09-29 Created: 2014-09-19 Last updated: 2014-09-30Bibliographically approved

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