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A note on unique games. (English) Zbl 1185.91017
Summary: We give a tighter analysis of the algorithm of S. Khot [“On the power of unique 2-prover 1-round games”, in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montreal, Quebec, Canada, 767–775 (2002)] which shows that given a unique 2-prover-1-round game with value $$1 - \varepsilon$$, one can find in polynomial time an assignment to the game with an expected weight of $$1 - O(k^{6/5} \varepsilon ^{1/5} (\log \frac {1}{\varepsilon k})^{2/5})$$, where $$k$$ is the size of the answer domain. This shows that if the unique games conjecture is true then the domain size $$k$$, must be at least $$\Omega ((\varepsilon ^{1/6}\log ^{1/3}(1/\varepsilon ))^{ - 1})$$, which is an improvement over the previous $$\Omega ((\varepsilon ^{1/10}\log ^{1/4}(1/\varepsilon ))^{ - 1})$$ bound.
##### MSC:
 91A05 2-person games 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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##### References:
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