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On the NP-hardness of approximating ordering constraint satisfaction problems
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0001-8217-0158
KTH, School of Computer Science and Communication (CSC).
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
2013 (English)In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, Springer, 2013, 26-41 p.Conference paper (Refereed)
Abstract [en]

We show improved NP-hardness of approximating Ordering Constraint Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of 14/15 + ε and 1/2 + ε. An OCSP is said to be approximation resistant if it is hard to approximate better than taking a uniformly random ordering. We prove that the Maximum Non- Betweenness Problem is approximation resistant and that there are width-m approximation-resistant OCSPs accepting only a fraction 1/(m/2)! of assignments. These results provide the first examples of approximation-resistant OCSPs subject only to P ≠ NP. Our reductions from Label Cover differ from previous works in two ways. First, we establish a somewhat general bucketing lemma permitting us to reduce the analysis of ordering predicates to that of classical predicates. Second, instead of "folding", which is not available for ordering predicates, we employ permuted instantiations of the predicates to limit the value of poorly correlated strategies.

Place, publisher, year, edition, pages
Springer, 2013. 26-41 p.
, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), ISSN 0302-9743 ; 8096 LNCS
Keyword [en]
Acyclic subgraph, Betweenness, Inapproximability, IS approximation, NP-hardness, Ordering constraints, Two ways
National Category
Computer and Information Science
URN: urn:nbn:se:kth:diva-136123DOI: 10.1007/978-3-642-40328-6_3ScopusID: 2-s2.0-84885206841ISBN: 978-364240327-9OAI: diva2:675596
16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013; Berkeley, CA; United States; 21 August 2013 through 23 August 2013

QC 20131204

Available from: 2013-12-04 Created: 2013-12-03 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.
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
urn:nbn:se:kth:diva-151402 (URN)
Public defence
2014-10-21, F3, Sing Sing, Lindstedtsvägen 26, KTH, Stockholm, 13:15 (English)
Approximation of NP-hard optimization problems
EU, European Research Council, 226203

QC 20140929

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

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