Efficient probabilistically checkable proofs and applications to approximations
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Two prover protocols: low error at affordable rates
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Improved non-approximability results
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Probabilistic checking of proofs and hardness of approximation problems
Probabilistic checking of proofs and hardness of approximation problems
Randomized graph products, chromatic numbers, and Lovasz j-function
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Interactive proofs and the hardness of approximating cliques
Journal of the ACM (JACM)
Testing of the long code and hardness for clique
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
On Unapproximable Versions of NP-Complete Problems
SIAM Journal on Computing
Some optimal inapproximability results
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
The value of strong inapproximability results for clique
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
A PCP characterization of NP with optimal amortized query complexity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Non-approximability results for optimization problems on bounded degree instances
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Zero Knowledge and the Chromatic Number
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
Hardness of Approximating the Minimum Distance of a Linear Code
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Simple Analysis of Graph Tests for Linearity and PCP
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Clique is hard to approximate within n1-
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Approximating Maximum Clique by Removing Subgraphs
SIAM Journal on Discrete Mathematics
The NP-completeness column: The many limits on approximation
ACM Transactions on Algorithms (TALG)
Finding large cliques in sparse semi-random graphs by simple randomized search heuristics
Theoretical Computer Science
On the approximability of the Maximum Agreement SubTree and Maximum Compatible Tree problems
Discrete Applied Mathematics
CSP gaps and reductions in the lasserre hierarchy
Proceedings of the forty-first annual ACM symposium on Theory of computing
Amplifying lower bounds by means of self-reducibility
Journal of the ACM (JACM)
2log1-ε n hardness for the closest vector problem with preprocessing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Streaming and communication complexity of clique approximation
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
On approximating string selection problems with outliers
CPM'12 Proceedings of the 23rd Annual conference on Combinatorial Pattern Matching
Improved exact algorithms for counting 3- and 4-colorings
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
On approximating string selection problems with outliers
Theoretical Computer Science
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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 $n/2^{(\log n)^{3/4+\gamma}}$ in an n vertex graph, assuming ${\rm NP} \nsubseteq {\rm BPTIME}(2^{(\log n)^{O(1)}})$. This improves the hardness factor of $n/2^{(\log n)^{1-\gamma'}}$ for some small (unspecified) constant γ′ 0 shown by Khot [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 $2^{\Omega(\sqrt{\log n})}$ for this problem, improving upon the hardness factor of (logn)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].