Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion

@inproceedings{Khot2006BetterIR,
  title={Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion},
  author={Subhash Khot and Ashok Kumar Ponnuswami},
  booktitle={ICALP},
  year={2006}
}
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… 

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References

SHOWING 1-10 OF 32 REFERENCES

Improved inapproximability results for MaxClique, chromatic number and approximate graph coloring

  • Subhash Khot
  • Computer Science
    Proceedings 2001 IEEE International Conference on Cluster Computing
  • 2001
The author presents improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic number of a graph, and the

On Unapproximable Versions of NP-Complete Problems

  • D. Zuckerman
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1996
TLDR
All of Karp's 21 original $NP$-complete problems have a version that is hard to approximate, and it is shown that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2SAT formula and computing the permanent of $-1, $0, $1$ matrices.

The value of strong inapproximability results for clique

TLDR
It is shown that for some such clique approximation problems that seem likely to require superpolynomial time in view of the results of Engebretsen and Holmerin, even certain low-degree polynomial lower bounds on their complexity will prove that N P ~ P.

ALGORITHMS FOR APPROXIMATE GRAPH COLORING

A coloring of a graph is an assignment of colors to the vertices so that no two adjacent vertices are given the same color. The problem of coloring a graph with the minimum number of colors is well

Randomized graph products, chromatic numbers, and the Lovász ϑ-function

TLDR
It is proved that for somec>0, there exists an infinite family of graphs such that vartheta (G) > α (G), n/2 c-sqrt {\log n} , wheren denotes the number of vertices in a graph.

Zero knowledge and the chromatic number

  • U. FeigeJ. Kilian
  • Mathematics, Computer Science
    Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)
  • 1996
TLDR
A new technique, inspired by zero-knowledge proof systems, is presented for proving lower bounds on approximating the chromatic number of a graph, and the result matches (up to low order terms) the known gap for approximation the size of the largest independent set.

Randomized graph products, chromatic numbers, and Lovasz j-function

TLDR
It is proved that for somec>0, there exists an infinite family of graphs such that vartheta (G) > α (G), where n denotes the number of vertices in a graph and this disproves a known conjecture regarding the ϑ function.

Hardness of approximation of the Balanced Complete Bipartite Subgraph problem

We prove that the Maximum Balanced Complete Bipartite Subgraph (BCBS) problem is hard to approximate within a factor of 2 n) δ for some δ > 0 under the plausible assumption that 3-SAT ∈ DTIME ( 2

Improved non-approximability results

TLDR
Strong non-approximability factors for central problems are indicated: N{Sup 1/4} for Max Clique; N{sup 1/10} for Chromatic Number; and 66/65 for Max 3SAT.