Towards optimal lower bounds for clique and chromatic number
It was previously known that neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n1−ε, for any constant ε>0, unless NP=ZPP. In this paper, we extend the reductions used to prove these results and combine the extended reductions with a recent result of Samorod...
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| Veröffentlicht in: | Theoretical computer science 2003-04, Vol.299 (1-3), p.537-584 |
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| creator | Engebretsen, Lars Holmerin, Jonas |
| description | It was previously known that neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n1−ε, for any constant ε>0, unless NP=ZPP. In this paper, we extend the reductions used to prove these results and combine the extended reductions with a recent result of Samorodnitsky and Trevisan to show that unless NP⊆ZPTIME(2O(logn(loglogn)3/2)), neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n1−ε(n) where ε∈O((loglogn)−1/2). Since there exists polynomial time algorithms approximating both problems within n1−ε(n) where ε(n)∈Ω(loglogn/logn), our result shows that the best possible ratio we can hope for is of the form n1−o(1), for some—yet unknown—value of o(1) between O((loglogn)−1/2) and Ω(loglogn/logn). |
| doi_str_mv | 10.1016/S0304-3975(02)00535-2 |
| format | Article |
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| source | ScienceDirect Journals (5 years ago - present); Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Algorithmics. Computability. Computer arithmetics Applied sciences Combinatorics Combinatorics. Ordered structures Computer science control theory systems Exact sciences and technology Graph theory Information retrieval. Graph Mathematics Operational research and scientific management Operational research. Management science Optimization. Search problems Sciences and techniques of general use Theoretical computing |
| title | Towards optimal lower bounds for clique and chromatic number |
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