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Parity is Positively Useless
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
2014 (English)In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: The 17th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems / [ed] Klaus Jansen, José Rolim, Nikhil Devanur, and Cristopher Moore, Dagstuhl, Germany: Schloss Dagstuhl , 2014, 433-448 p.Conference paper, Published paper (Refereed)
Abstract [en]

We give the first examples of non-trivially positively-useless predicates subject only to P != NP. In particular, for every constraint function Q : {-1,1}^4 -> R, we construct Contraint-Satisfaction-Problem (CSP) instances without negations which have value at least 1-eps when evaluted for the arity-four odd-parity predicate, yet it is NP-hard to find a solution with value significantly better than a random biased assignment when evaluated for Q. More generally, we show that all parities except one are positively useless. Although we are not able to exhibit a single protocol producing hard instances when evaluated for every Q, we show that two protocols do the trick. The first protocol is the classical one used by Håstad with a twist. We extend the protocol to multilayered Label Cover and employ a particular distribution over layers in order to limit moments of table biases. The second protocol is a modification of Chan's multi-question protocol where queried tuples of Label Cover vertices are randomized in such a way that the tables can be seen as being independently sampled from a common distribution and in effect having identical expected biases. We believe that our techniques may prove useful in further analyzing the approximability of CSPs without negations.

Place, publisher, year, edition, pages
Dagstuhl, Germany: Schloss Dagstuhl , 2014. 433-448 p.
Series
APPROX, ISSN 1868-8969 ; 17
Keyword [en]
Approximation hardness, approximation resistance, parity, usefulness, negations, monotone, constraint satisfaction problems, smoothness, multilayer
National Category
Computer Science
Research subject
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-151401DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.433Scopus ID: 2-s2.0-84920192892ISBN: 978-3-939897-74-3 (print)OAI: oai:DiVA.org:kth-151401DiVA: diva2:748473
Conference
APPROX/RANDOM 2014
Projects
Approximation of NP-hard optimization problems
Funder
EU, European Research Council, 226203
Note

QC 20140922

Available from: 2014-09-19 Created: 2014-09-19 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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