TY - JOUR

T1 - Approximating minimum power covers of intersecting families and directed edge-connectivity problems

AU - Nutov, Zeev

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010/6/6

Y1 - 2010/6/6

N2 - Given a (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 the graph is the sum of the powers of its nodes. Let G = (V, E) be a graph with edge costs {c (e) : e ∈ E} and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Powerk-Edge-Outconnected Subgraph problem we are given a root r ∈ V, and require that G contains k pairwise edge-disjoint r v-paths for all v ∈ V - r. In the Min-Powerk-Edge-Connected Subgraph problem G is required to be k-edge-connected. For k = 1, these problems are at least as hard as the Set-Cover problem and thus have an Ω (ln | V |) approximation threshold. For k = Ω (nε), they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O (k ln | V |). Our algorithms are based on a more general O (ln | V |)-approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F is intersecting if X ∩ Y, X ∪ Y ∈ F for any intersecting X, Y ∈ F, and an edge set I covers F if for every X ∈ F there is an edge in I entering X.

AB - Given a (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 the graph is the sum of the powers of its nodes. Let G = (V, E) be a graph with edge costs {c (e) : e ∈ E} and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Powerk-Edge-Outconnected Subgraph problem we are given a root r ∈ V, and require that G contains k pairwise edge-disjoint r v-paths for all v ∈ V - r. In the Min-Powerk-Edge-Connected Subgraph problem G is required to be k-edge-connected. For k = 1, these problems are at least as hard as the Set-Cover problem and thus have an Ω (ln | V |) approximation threshold. For k = Ω (nε), they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O (k ln | V |). Our algorithms are based on a more general O (ln | V |)-approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F is intersecting if X ∩ Y, X ∪ Y ∈ F for any intersecting X, Y ∈ F, and an edge set I covers F if for every X ∈ F there is an edge in I entering X.

KW - Approximation algorithms

KW - Directed graphs

KW - Edge-connectivity

KW - Intersecting families

KW - Power minimization

KW - Wireless networks

UR - http://www.scopus.com/inward/record.url?scp=77953284882&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2010.03.009

DO - 10.1016/j.tcs.2010.03.009

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AN - SCOPUS:77953284882

SN - 0304-3975

VL - 411

SP - 2502

EP - 2512

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 26-28

ER -