Streaming Algorithms for Estimating the Matching Size in Planar Graphs and Beyond

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. When the input graph is planar, we present a simple and elegant streaming algorithm that with high probability estimates the size of a maximum matching within a constant factor using [EQUATION](n2/3) space, where n is the number of vertices. The approach generalizes to the family of graphs that have bounded arboricity, which include graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model) in o(n) space. We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We further reduce the required memory size to [EQUATION] for three restricted settings: (i) when the input graph is a forest; (ii) when we have 2-passes and the input graph has bounded arboricity; and (iii) when the edges arrive in random order and the input graph has bounded arboricity. Finally, we design a reduction from the Boolean Hidden Matching Problem to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o(n1/2) bits of space. Using the same reduction, we show that there is no deterministic algorithm that computes this kind of estimate in o(n) bits of space. The lower bounds hold even for graphs that are collections of paths of constant length.

[1]  A. Kemper,et al.  On Graph Problems in a Semi-streaming Model , 2015 .

[2]  Ashish Goel,et al.  On the communication and streaming complexity of maximum bipartite matching , 2012, SODA.

[3]  Krzysztof Onak,et al.  Maintaining a large matching and a small vertex cover , 2010, STOC '10.

[4]  Yuichi Yoshida,et al.  An improved constant-time approximation algorithm for maximum~matchings , 2009, STOC '09.

[5]  Krzysztof Onak,et al.  Local Graph Partitions for Approximation and Testing , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[6]  Sanjeev Khanna,et al.  Approximating matching size from random streams , 2014, SODA.

[7]  Anand Srivastav,et al.  Bipartite Graph Matchings in the Semi-streaming Model , 2009, ESA.

[8]  Joan Feigenbaum,et al.  Graph Distances in the Data-Stream Model , 2008, SIAM J. Comput..

[9]  Mikhail Kapralov,et al.  Better bounds for matchings in the streaming model , 2012, SODA.

[10]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[11]  Silvio Micali,et al.  An O(v|v| c |E|) algoithm for finding maximum matching in general graphs , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[12]  Wei Yu,et al.  The streaming complexity of cycle counting, sorting by reversals, and other problems , 2011, SODA '11.

[13]  Shay Solomon,et al.  Simple deterministic algorithms for fully dynamic maximal matching , 2012, STOC '13.

[14]  Yijie Han.  Matching for Graphs of Bounded Degree , 2008, FAW.

[15]  Venkatesan Guruswami,et al.  Superlinear Lower Bounds for Multipass Graph Processing , 2012, Algorithmica.

[16]  Richard M. Karp,et al.  An optimal algorithm for on-line bipartite matching , 1990, STOC '90.

[17]  Krzysztof Onak,et al.  Constant-Time Approximation Algorithms via Local Improvements , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[18]  Ran Raz,et al.  Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography , 2008, SIAM J. Comput..

[19]  Richard Peng,et al.  Fully Dynamic $(1+\epsilon)$-Approximate Matchings , 2013, 1304.0378.

[20]  Ziv Bar-Yossef,et al.  Exponential separation of quantum and classical one-way communication complexity , 2004, STOC '04.

[21]  Aravind Srinivasan,et al.  Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds , 1997, SIAM J. Comput..

[22]  Leah Epstein,et al.  Improved Approximation Guarantees for Weighted Matching in the Semi-streaming Model , 2009, SIAM J. Discret. Math..

[23]  Sandeep Sen,et al.  Fully Dynamic Maximal Matching in O (log n) Update Time , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[24]  Krzysztof Onak,et al.  A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size , 2011, SODA.

[25]  Alexandr V. Kostochka,et al.  Lower bound of the hadwiger number of graphs by their average degree , 1984, Comb..

[26]  L. Lovász.  Matching Theory (North-Holland mathematics studies) , 1986 .

[27]  Errol L. Lloyd,et al.  Fully Dynamic Maintenance of Vertex Cover , 1993, WG.

[28]  Andrew McGregor,et al.  Finding Graph Matchings in Data Streams , 2005, APPROX-RANDOM.

[29]  List of Open Problems in Sublinear Algorithms , .

[30]  Anand Srivastav,et al.  Bipartite Matching in the Semi-streaming Model , 2012, Algorithmica.

[31]  Dana Ron,et al.  On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms , 2007, Electron. Colloquium Comput. Complex..

[32]  Claire Mathieu,et al.  Maximum Matching in Semi-streaming with Few Passes , 2011, APPROX-RANDOM.

[33]  Mariano Zelke,et al.  Weighted Matching in the Semi-Streaming Model , 2007, Algorithmica.