Better inapproximability results for maxclique, chromatic number and Min-3Lin-Deletion

Subhash Khot, Ashok Kumar Ponnuswami

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph and the problem of finding the chromatic number of a graph. We show that for any constant γ > 0, there is no polynomial time algorithm that approximates these problems within factor n/2(log n)3/4+γ in an n vertex graph, assuming NP ⊈ BPTIME(2(log n)o(1)). This improves the hardness factor of n/2(log n)1-γ′ for some small (unspecified) constant γ′ > 0 shown by Knot [20]. Our main idea is to show an improved hardness result for the Min-3Lin-Deletion problem. An instance of Min-3Lin-Deletion is a system of linear equations modulo 2, where each equation is over three variables. The objective is to find the minimum number of equations that need to be deleted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of 2 Ω(√log n) for this problem, improving upon the hardness factor of (log n)β shown by Håstad [18], for some small (unspecified) constant β > 0. The hardness results for clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given in [20].

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings
PublisherSpringer Verlag
Number of pages12
ISBN (Print)3540359044, 9783540359043
StatePublished - 2006
Event33rd International Colloquium on Automata, Languages and Programming, ICALP 2006 - Venice, Italy
Duration: Jul 10 2006Jul 14 2006

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4051 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other33rd International Colloquium on Automata, Languages and Programming, ICALP 2006

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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