Chapter

LATIN 2008: Theoretical Informatics

Volume 4957 of the series Lecture Notes in Computer Science pp 423-435

Approximating Minimum-Power Degree and Connectivity Problems

  • Guy KortsarzAffiliated withRutgers University, Camden
  • , Vahab S. MirrokniAffiliated withMicrosoft Research
  • , Zeev NutovAffiliated withThe Open University of Israel, Raanana
  • , Elena TsankoAffiliated withIBM, Haifa

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Abstract

Power optimization is a central issue in wireless network design. Given a (possibly directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected network design problems under the power minimization criteria. Given a graph https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq1_HTML.png with edge costs https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq2_HTML.png and degree requirements {r(v):v ∈ V}, the Minimum-Power Edge-Multi-Cover ( https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq3_HTML.png ) problem is to find a minimum-power subgraph of https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq4_HTML.png so that the degree of every node v is at least r(v). We give an O(logn)-approximation algorithms for https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq5_HTML.png , improving the previous ratio O(log4 n) of [11]. This is used to derive an O(logn + α)-approximation algorithm for the undirected Minimum-Power k -Connected Subgraph ( https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq6_HTML.png ) problem, where https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq7_HTML.png is the best known ratio for the min-cost variant of the problem (currently, https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq8_HTML.png for n ≥ 2k 2 and https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq9_HTML.png otherwise). Surprisingly, it shows that the min-power and the min-cost versions of the k -Connected Subgraph problem are equivalent with respect to approximation, unless the min-cost variant admits an o(logn)-approximation, which seems to be out of reach at the moment. We also improve the best known approximation ratios for small requirements. Specifically, we give a 3/2-approximation algorithm for https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq10_HTML.png with r(v) ∈ {0,1}, improving over the 2-approximation by [11], and a \(3\frac{2}{3}\)-approximation for the minimum-power 2-Connected and 2-Edge-Connected Subgraph problems, improving the 4-approximation by [4]. Finally, we give a 4 r max -approximation algorithm for the undirected Minimum-Power Steiner Network ( https://static-content.springer.com/image/chp%3A10.1007%2F978-3-540-78773-0_37/MediaObjects/978-3-540-78773-0_37_IEq12_HTML.png ) problem: find a minimum-power subgraph that contains r(u,v) pairwise edge-disjoint paths for every pair u,v of nodes.