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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Veröffentlicht in:Algorithmica Jg. 63; H. 1-2; S. 490 - 508
Hauptverfasser: Eggert, Sebastian, Kliemann, Lasse, Munstermann, Peter, Srivastav, Anand
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer-Verlag 01.06.2012
Springer
Schlagworte:
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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Beschreibung
Zusammenfassung: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.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-011-9556-8