On the hardness of approximating label-cover

Irit Dinur*, Shmuel Safra

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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(logn)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.

Original languageEnglish
Pages (from-to)247-254
Number of pages8
JournalInformation Processing Letters
Issue number5
StatePublished - 16 Mar 2004


  • Computational complexity
  • Hardness of approximation
  • Label-cover
  • PCP


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