A Pseudo-Deterministic RNC Algorithm for General Graph Perfect Matching
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The difficulty of obtaining an NC perfect matching algorithm has led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently [GG15] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same unique perfect matching for almost all choices of random bits. In this paper, we give an analogous algorithm for general, not necessarily bipartite, graphs. More generally, we give a pseudo-deterministic RNC algorithm for finding a minimum weight perfect matching when the edge weights are polynomially bounded. Our algorithm builds on [AV18], whose result used planarity of input graphs critically; in fact, in three different ways. The main challenge was to adapt these steps to general graphs by exploiting the leeway that seeking a pseudo-deterministic RNC algorithm and not an NC algorithm gives us.
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