Approximation and inapproximability results for maximum clique of disc graphs in high dimensions

Authors:
Peyman Afshani;Hamed Hatami
Affiliations:
School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada;Department of Computer Science, University of Toronto, Toronto, Ontario, Canada
Venue:
Information Processing Letters
Year:
2008

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Abstract

We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if A^* is the largest subset of diameter r of n points in the Euclidean space, then for every @e0 there exists a polynomial time algorithm that outputs a set B of size at least |A^*| and of diameter at most r(2+@e). On the hardness side, roughly speaking, we show that unless P=NP for every @e0 it is not possible to guarantee the diameter r(4/3-@e) for B even if the algorithm is allowed to output a set of size (9594-@e)^-^1|A^*|.