Moderately exponential time algorithms for the maximum induced matching problem View Full Text


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Article Info

DATE

2014-10-22

AUTHORS

Maw-Shang Chang, Li-Hsuan Chen, Ling-Ju Hung

ABSTRACT

An induced matchingM⊆E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\subseteq E$$\end{document} in a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V, E)$$\end{document} is a matching such that no two edges in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document} are joined by any third edge of the graph. The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. It is NP-hard. Branch-and-reduce algorithms proposed in the previous results for the Maximum Induced Matching problem use a standard branching rule: for a vertex v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document}, it branches into deg(v)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$deg(v)+1$$\end{document} subproblems that either v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} is not an endvertex of any edge in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document} or v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} and one of its neighbor are endvertices of an edge in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}. In this paper, we give a simple branch-and-reduce algorithm consisting of a boundary condition, a reduction rule, and a branching rule. Especially, the branching rule only branches the original problem into two subproblems. When the algorithm meets the input instance satisfying the boundary condition, we reduce the Maximum Induced Matching problem to the Maximum Independent Set problem. By using the measure-and-conquer approach to analyze the running time of the algorithm, we show that this algorithm runs in time O∗(1.4658n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^{*}(1.4658^n)$$\end{document} which is faster than previously known algorithms. By adding two branching rules in the simple exact algorithm, we obtain an O∗(1.4321n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^{*}(1.4321^n)$$\end{document}-time algorithm for the Maximum Induced Matching problem. Moreover, we give a moderately exponential time ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-approximation algorithm, 0<ρ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \rho < 1$$\end{document}, for the Maximum Induced Matching problem. For ρ=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho =0.5$$\end{document}, the algorithm runs in time O∗(1.3348n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^{*}(1.3348^n)$$\end{document}. More... »

PAGES

981-998

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11590-014-0813-z

DOI

http://dx.doi.org/10.1007/s11590-014-0813-z

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1008036174


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In this paper, we give a simple branch-and-reduce algorithm consisting of a boundary condition, a reduction rule, and a branching rule. Especially, the branching rule only branches the original problem into two subproblems. When the algorithm meets the input instance satisfying the boundary condition, we reduce the Maximum Induced Matching problem to the Maximum Independent Set problem. By using the measure-and-conquer approach to analyze the running time of the algorithm, we show that this algorithm runs in time O∗(1.4658n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^{*}(1.4658^n)$$\end{document} which is faster than previously known algorithms. By adding two branching rules in the simple exact algorithm, we obtain an O∗(1.4321n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^{*}(1.4321^n)$$\end{document}-time algorithm for the Maximum Induced Matching problem. Moreover, we give a moderately exponential time ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-approximation algorithm, 0<ρ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \rho < 1$$\end{document}, for the Maximum Induced Matching problem. For ρ=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho =0.5$$\end{document}, the algorithm runs in time O∗(1.3348n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^{*}(1.3348^n)$$\end{document}.
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18 schema:keywords Induced Matching problem
19 Maximum Induced Matching problem
20 NP
21 algorithm
22 approach
23 boundary conditions
24 branches
25 branching rules
26 cardinality
27 conditions
28 conquer approach
29 edge
30 endvertex
31 endvertices
32 exact algorithm
33 exponential time
34 exponential time algorithm
35 graph
36 independent set problem
37 induced matching
38 input instances
39 instances
40 matching
41 matching problem
42 maximum cardinality
43 maximum independent set problem
44 measures
45 neighbors
46 original problem
47 paper
48 previous results
49 problem
50 reduction rules
51 results
52 rules
53 running time
54 set problem
55 simple branch
56 simple exact algorithm
57 standard branching rule
58 subproblems
59 third edge
60 time
61 time algorithm
62 vertices
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