Maximum Matching in Two, Three, and a Few More Passes over Graph Stream

We consider the maximum matching problem in the semi-streaming model formalized by Feigenbaum, Kannan, McGregor, Suri, and Zhang [13] that is inspired by giant graphs of today. As our main result, we give a two-pass (1/2 + 1/16)-approximation algorithm for triangle-free graphs and a two-pass (1/2 + 1/32)-approximation algorithm for general graphs; these improve the approximation ratios of 1/2 + 1/52 for bipartite graphs and 1/2 + 1/140 for general graphs by Konrad, Magniez, and Mathieu [20]. In three passes, we are able to achieve approximation ratios of 1/2 + 1/10 for triangle-free graphs and 1/2 + 1/19.753 for general graphs. We also give a multi-pass algorithm where we bound the number of passes precisely – we give a (2/3 − ε)approximation algorithm that uses 2/(3ε) passes for triangle-free graphs and 4/(3ε) passes for general graphs. Our algorithms are simple and combinatorial, use O(n logn) space, and (can be implemented to) have O(1) update time per edge. For general graphs, our multi-pass algorithm improves the best known deterministic algorithms in terms of the number of passes: Ahn and Guha [1] give a (2/3− ε)-approximation algorithm that uses O(log(1/ε)/ε2) passes, whereas our (2/3− ε)-approximation algorithm uses 4/(3ε) passes; they also give a (1−ε)-approximation algorithm that uses O(logn ·poly(1/ε)) passes, where n is the number of vertices of the input graph; although our algorithm is (2/3−ε)-approximation, our number of passes do not depend on n. Earlier multi-pass algorithms either have a large constant inside big-O notation for the number of passes [9] or the constant cannot be determined due to the involved analysis [22, 1], so our multi-pass algorithm should use much fewer passes for approximation ratios bounded slightly below 2/3. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems