Bipartite Matching in the Semi-streaming Model

We present the first deterministic 1+ ε approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O ((1/ ε ) 5 ) passes over the input stream. In this model, the input graph G =( V , E ) is given as a stream of its edges in some arbitra...

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Published in:	Algorithmica Vol. 63; no. 1-2; pp. 490 - 508
Main Authors:	Eggert, Sebastian, Kliemann, Lasse, Munstermann, Peter, Srivastav, Anand
Format:	Journal Article
Language:	English
Published:	New York Springer-Verlag 01.06.2012 Springer
Subjects:	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Combinatorics Combinatorics. Ordered structures Computer Science Computer science; control theory; systems Computer systems and distributed systems. User interface Computer Systems Organization and Communication Networks Data Structures and Information Theory Exact sciences and technology Graph theory Information retrieval. Graph Mathematics Mathematics of Computing Sciences and techniques of general use Software Theoretical computing Theory of Computation Streaming algorithms Approximation algorithms Bipartite graph matching Approximation schemes Streaming Combinatorial problem Graph theory Approximation algorithm Randomized algorithm Combinatorial optimization Modeling Backtracking Randomization Storage Randomized design Search tree Bipartite graph Deterministic approach Algorithm analysis Graph matching Novelty
ISSN:	0178-4617, 1432-0541
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Abstract	We present the first deterministic 1+ ε approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O ((1/ ε ) 5 ) passes over the input stream. In this model, the input graph G =( V , E ) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O ( n polylog n ) bits, where . The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005 ), which uses Ω ((1/ ε ) 1/ ε ) passes. We show that even for bipartite graphs, McGregor’s algorithm needs Ω (1/ ε ) Ω (1/ ε ) passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization. We repeatedly grow an initial matching using augmenting paths up to a length of 2 k +1 for k =⌈2/ ε ⌉. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k , that guarantees a 1+ ε approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits : when a matching edge is the i th matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i . This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand.
AbstractList	We present the first deterministic 1+ ε approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O ((1/ ε ) 5 ) passes over the input stream. In this model, the input graph G =( V , E ) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O ( n polylog n ) bits, where . The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005 ), which uses Ω ((1/ ε ) 1/ ε ) passes. We show that even for bipartite graphs, McGregor’s algorithm needs Ω (1/ ε ) Ω (1/ ε ) passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization. We repeatedly grow an initial matching using augmenting paths up to a length of 2 k +1 for k =⌈2/ ε ⌉. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k , that guarantees a 1+ ε approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits : when a matching edge is the i th matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i . This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand.
Author	Srivastav, Anand Eggert, Sebastian Kliemann, Lasse Munstermann, Peter
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Keywords	Streaming algorithms Approximation algorithms Bipartite graph matching Approximation schemes Streaming Combinatorial problem Graph theory Approximation algorithm Randomized algorithm Combinatorial optimization Modeling Backtracking Randomization Storage Randomized design Search tree Bipartite graph Deterministic approach Algorithm analysis Graph matching Novelty
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References_xml	– reference: EggertS.KliemannL.SrivastavA.Bipartite graph matchings in the semi-streaming modelProceedings of the 17th Annual European Symposium on Algorithms (ESA 2009)2009492503Presented also at the MADALGO Workshop on Massive Data Algorithmics, June 2009, Århus, Denmark. – reference: Comput. J. 51(1,3), (2008). Special issue on parameterized complexity. Available from: http://comjnl.oxfordjournals.org/content/51/1.toc – reference: FeigenbaumJ.KannanS.McGregorA.SuriS.ZhangJ.On graph problems in a semi-streaming modelProceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004)200453154310.1007/978-3-540-27836-8_46 – reference: KliemannL.Matching in bipartite graph streams in a small number of passes (extended abstract)Proceedings of the 10th International Symposium on Experimental and Efficient Algorithms (SEA 2011)2011254266Document ID: dda51148-ac5b-4655-9c4f-e01f26511235 – reference: MuchaM.SankowskiP.Maximum matchings via Gaussian eliminationProceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, October 2004 (FOCS 2004)2004248255Available from: http://www.mimuw.edu.pl/~mucha/pub/mucha_sankowski_focs04.pdf. Available from: http://doi.ieeecomputersociety.org/10.1109/FOCS.2004.40 – reference: McGregorA.Finding graph matchings in data streamsProceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation (APPROX RANDOM 2005)2005170181 – reference: Ahn, K.J., Guha, S.: Linear programming in the semi-streaming model with application to the maximum matching problem (2011). A short version appeared at ICALP 2011, pp. 526–538. Available from: http://arxiv.org/abs/1104.2315 – reference: BergeC.Two theorems in graph theoryProc. Natl. Acad. Sci. USA1957439842844948110086.1620210.1073/pnas.43.9.842Available from: http://www.pnas.org/content/43/9/842.short – reference: MuthukrishnanS.M.Data streams: algorithms and applicationsFound. Trends Theor. Comput. Sci.200512672379507Available from: http://algo.research.googlepages.com/eight.ps – reference: HopcroftJ.E.KarpR.M.An n5/2 algorithm for maximum matchings in bipartite graphsSIAM J. Comput.1973242252313376990266.0511410.1137/0202019 – start-page: 248 volume-title: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, October 2004 (FOCS 2004) year: 2004 ident: 9556_CR9 – start-page: 492 volume-title: Proceedings of the 17th Annual European Symposium on Algorithms (ESA 2009) year: 2009 ident: 9556_CR4 – start-page: 254 volume-title: Proceedings of the 10th International Symposium on Experimental and Efficient Algorithms (SEA 2011) year: 2011 ident: 9556_CR7 – volume: 43 start-page: 842 issue: 9 year: 1957 ident: 9556_CR2 publication-title: Proc. Natl. Acad. Sci. USA doi: 10.1073/pnas.43.9.842 – volume: 2 start-page: 225 issue: 4 year: 1973 ident: 9556_CR6 publication-title: SIAM J. Comput. doi: 10.1137/0202019 – volume: 1 start-page: 67 issue: 2 year: 2005 ident: 9556_CR10 publication-title: Found. Trends Theor. Comput. Sci. doi: 10.1561/0400000002 – ident: 9556_CR1 – start-page: 531 volume-title: Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004) year: 2004 ident: 9556_CR5 doi: 10.1007/978-3-540-27836-8_46 – ident: 9556_CR3 – start-page: 170 volume-title: Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation (APPROX RANDOM 2005) year: 2005 ident: 9556_CR8 doi: 10.1007/11538462_15
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Combinatorics Combinatorics. Ordered structures Computer Science Computer science; control theory; systems Computer systems and distributed systems. User interface Computer Systems Organization and Communication Networks Data Structures and Information Theory Exact sciences and technology Graph theory Information retrieval. Graph Mathematics Mathematics of Computing Sciences and techniques of general use Software Theoretical computing Theory of Computation
Title	Bipartite Matching in the Semi-streaming Model
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