Optimal Streaming Algorithms for Graph Matching
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We present parameterized streaming algorithms for the graph matching problem in both the dynamic and the insert-only models. For the dynamic streaming model, we present a one-pass algorithm that, with high probability, computes a maximum-weight k-matching of a weighted graph in Õ(Wk^2) space and that has Õ(1) update time, where W is the number of distinct edge weights and the notation Õ() hides a poly-logarithmic factor in the input size. For the insert-only streaming model, we present a one-pass algorithm that runs in O(k^2) space and has O(1) update time, and that, with high probability, computes a maximum-weight k-matching of a weighted graph. The space complexity and the update-time complexity achieved by our algorithms for unweighted k-matching in the dynamic model and for weighted k-matching in the insert-only model are optimal. A notable contribution of this paper is that the presented algorithms do not rely on the apriori knowledge/promise that the cardinality of every maximum-weight matching of the input graph is upper bounded by the parameter k. This promise has been a critical condition in previous works, and lifting it required the development of new tools and techniques.
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