Given a set of points Q in the plane, define the r/2-Disk Graph, Q(r), as a generalized version of the Unit Disk Graph: the vertices of the graph is Q and there is an edge between two points in Q iff the distance between them is at most r. In this paper, motivated by applications in wireless sensor networks, we study the following geometric problem of color-spanning sets: given n points with m colors in the plane, choosing m points P with distinct colors such that the r/2-Disk Graph, P(r), is connected and r is minimized. When at most two points are of the same color ci (or, equivalently, when a color ci spans at most two points), we prove that the problem is NP-hard to approximate within a factor 3 - ε. And we present a tight factor-3 approximation for this problem. For the more general case when each color spans at most k points, we present a factor-(2k-1) approximation. Our solutions are based on the applications of the famous Hall's Marriage Theorem on bipartite graphs, which could be useful for other problems. © 2013 Springer-Verlag.
CITATION STYLE
Fan, C., Luo, J., & Zhu, B. (2013). Tight approximation bounds for connectivity with a color-spanning set. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8283 LNCS, pp. 590–600). https://doi.org/10.1007/978-3-642-45030-3_55
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