On the hardness of approximating label-cover

The Label-Cover problem, defined by S. Arora, L. Babai, J. Stern, Z. Sweedyk [Proceedings of 34th IEEE Symposium on Foundations of Computer Science, 1993, pp. 724–733], serves as a starting point for numerous hardness of approximation reductions. It is one of six ‘canonical’ approximation problems i...

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Bibliographic Details
Published in:Information processing letters 2004-03, Vol.89 (5), p.247-254
Main Authors: Dinur, Irit, Safra, Shmuel
Format: Article
Language:English
Subjects:
Applied sciences
Approximation
Computational complexity
Computer science
Computer science
control theory
systems
Estimates
Exact sciences and technology
Hardness of approximation
Label-cover
Miscellaneous
PCP
Studies
Theoretical computing
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Summary:The Label-Cover problem, defined by S. Arora, L. Babai, J. Stern, Z. Sweedyk [Proceedings of 34th IEEE Symposium on Foundations of Computer Science, 1993, pp. 724–733], serves as a starting point for numerous hardness of approximation reductions. It is one of six ‘canonical’ approximation problems in the survey of Arora and Lund [Hardness of Approximations, in: Approximation Algorithms for NP-Hard Problems, PWS Publishing Company, 1996, Chapter 10]. In this paper we present a direct combinatorial reduction from low error-probability PCP [Proceedings of 31st ACM Symposium on Theory of Computing, 1999, pp. 29–40] to Label-Cover showing it NP-hard to approximate to within 2 (log n) 1−o(1) . This improves upon the best previous hardness of approximation results known for this problem. We also consider the Minimum-Monotone-Satisfying-Assignment (MMSA) problem of finding a satisfying assignment to a monotone formula with the least number of 1's, introduced by M. Alekhnovich, S. Buss, S. Moran, T. Pitassi [Minimum propositional proof length is NP-hard to linearly approximate, 1998]. We define a hierarchy of approximation problems obtained by restricting the number of alternations of the monotone formula. This hierarchy turns out to be equivalent to an AND/OR scheduling hierarchy suggested by M.H. Goldwasser, R. Motwani [Lecture Notes in Comput. Sci., Vol. 1272, Springer-Verlag, 1997, pp. 307–320]. We show some hardness results for certain levels in this hierarchy, and place Label-Cover between levels 3 and 4. This partially answers an open problem from M.H. Goldwasser, R. Motwani regarding the precise complexity of each level in the hierarchy, and the place of Label-Cover in it.
ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2003.11.007