Power Optimization for Connectivity Problems

Given a 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 the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimi...

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Bibliographic Details
Published inInteger Programming and Combinatorial Optimization pp. 349 - 361
Main Authors Hajiaghayi, Mohammad T., Kortsarz, Guy, Mirrokni, Vahab S., Nutov, Zeev
Format Book Chapter Conference Proceeding
LanguageEnglish
Published Berlin, Heidelberg Springer Berlin Heidelberg 2005
Springer
SeriesLecture Notes in Computer Science
Subjects
Applied sciences
Approximation Algorithm
Approximation Ratio
Connectivity Problem
Exact sciences and technology
Flows in networks. Combinatorial problems
Operational research and scientific management
Operational research. Management science
Power Optimization
Span Subgraph
Graph path
Graph node
Wireless telecommunication
Disjoint edge
Minimization
Approximation algorithm
Combinatorial optimization
Polynomial time
Direct problem
Integer programming
Graph connectivity
NP hard problem
Telecommunication network
Subgraph
Directed graph
Threshold
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ISBN9783540261995
3540261990
ISSN0302-9743
1611-3349
DOI10.1007/11496915_26

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Summary:Given a 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 the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-ECSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log4n)-approximation algorithm for MPb-EC. This gives an O(log4n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an $O(\sqrt{n})$ approximation algorithm for MPk-ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2log1 − εn) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time.
Bibliography:Original Abstract: Given a 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 the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-ECSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log4n)-approximation algorithm for MPb-EC. This gives an O(log4n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\sqrt{n})$\end{document} approximation algorithm for MPk-ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2log1 − εn) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time.
ISBN:9783540261995
3540261990
ISSN:0302-9743
1611-3349
DOI:10.1007/11496915_26