Deterministic Fully Dynamic Approximate Vertex Cover and Fractional Matching in O(1) Amortized Update Time

We consider the problems of maintaining approximate maximum matching and minimum vertex cover in a dynamic graph. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupt...

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Published inInteger Programming and Combinatorial Optimization Vol. 10328; pp. 86 - 98
Main Authors Bhattacharya, Sayan, Chakrabarty, Deeparnab, Henzinger, Monika
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 01.01.2017
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Approximation Ratio
Dynamic Algorithm
Input Graph
Maximum Match
Vertex Cover
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ISBN9783319592497
3319592491
ISSN0302-9743
1611-3349
DOI10.1007/978-3-319-59250-3_8

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Summary:We consider the problems of maintaining approximate maximum matching and minimum vertex cover in a dynamic graph. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized 2-approximation dynamic algorithm for this problem that has amortized update time of O(1) with high probability. We consider the natural open question of derandomizing this result. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of $$(2+\epsilon )$$ and an amortized update time of $$O(\log n/\epsilon ^2)$$ . Our result can be generalized to give a fully dynamic $$O(f^3)$$ -approximation algorithm with $$O(f^2)$$ amortized update time for the hypergraph vertex cover and fractional matching problems, where every hyperedge has at most f vertices.
Bibliography:D. Chakrabarty—Work done while the author was at Microsoft Research, India. M. Henzinger—The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement number 340506.
Original Abstract: We consider the problems of maintaining approximate maximum matching and minimum vertex cover in a dynamic graph. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized 2-approximation dynamic algorithm for this problem that has amortized update time of O(1) with high probability. We consider the natural open question of derandomizing this result. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+\epsilon )$$\end{document} and an amortized update time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log n/\epsilon ^2)$$\end{document}. Our result can be generalized to give a fully dynamic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(f^3)$$\end{document}-approximation algorithm with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(f^2)$$\end{document} amortized update time for the hypergraph vertex cover and fractional matching problems, where every hyperedge has at most f vertices.
ISBN:9783319592497
3319592491
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-319-59250-3_8