Bipartite Matching in the Semi-streaming Model.

Eggert, Sebastian, Kliemann, Lasse, Munstermann, Peter and Srivastav, Anand (2012) Bipartite Matching in the Semi-streaming Model. Algorithmica, 63 (1-2). pp. 490-508. DOI 10.1007/s00453-011-9556-8.

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Abstract

We present the first deterministic 1+ε approximational gorithm for finding alarge 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(npolylogn) bits, where n=|V|. 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 2k+1 fork=�2/ε�. We terminate when the number of augmenting paths found inone 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 ith 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.

Document Type: Article
Research affiliation: Kiel University > Kiel Marine Science
OceanRep > The Future Ocean - Cluster of Excellence
Kiel University
Refereed: Yes
Open Access Journal?: No
DOI etc.: 10.1007/s00453-011-9556-8
ISSN: 0178-4617
Projects: Future Ocean
Date Deposited: 29 Mar 2018 10:16
Last Modified: 04 Apr 2019 08:39
URI: http://oceanrep.geomar.de/id/eprint/42521

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