Packing and covering δ-hyperbolic spaces by balls
نویسندگان
چکیده
We consider the problem of covering and packing subsets of δ-hyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δhyperbolic graph G and a positive number R, let γ(S, R) be the minimum number of balls of radius R covering S. It is known that computing γ(S, R) or approximating this number within a constant factor is hard even for 2-hyperbolic graphs. In this paper, using a primal-dual approach, we show how to construct in polynomial time a covering of S with at most γ(S, R) balls of (slightly larger) radius R + δ. This result is established in the general framework of δ-hyperbolic geodesic metric spaces and is extended to some other set families derived from balls. This covering algorithm is used to design better than in general case approximation algorithms for the augmentation problem of δ-hyperbolic graphs with diameter constraints and slackness δ and for the k-center problem in δ-hyperbolic graphs.
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