Maintaining Approximate Maximum Matching in an Incremental Bipartite Graph in Polylogarithmic Update Time

A sparse subgraph G' of G is called a matching sparsifier if the size or weight of matching in G' is approximately equal to the size or weight of maximum matching in G. Recently, algorithms have been developed to find matching sparsifiers in a static bipartite graph. In this paper, we show that we can find matching sparsifier even in an incremental bipartite graph. This observation leads to following results: 1. We design an algorithm that maintains a (1+epsilon) approximate matching in an incremental bipartite graph in O(log^2(n) / (epsilon^{4}) update time. 2. For weighted graphs, we design an algorithm that maintains (1+epsilon) approximate weighted matching in O((log(n)*log(n*N)) / (epsilon^4) update time where \maxweight is the maximum weight of any edge in the graph.

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