Matching is as Easy as the Decision Problem, in the NC Model
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We give an NC reduction from search to decision for the problem of finding a minimum weight perfect matching, provided edge weights are polynomially bounded. As a consequence, for settling the long-standing open problem of obtaining an NC perfect matching algorithm, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. The difficulty of obtaining an NC perfect matching algorithm 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 (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs. An equivalent way of stating our main result is: We give an NC algorithm for finding a minimum weight perfect matching in a general graph with polynomially bounded edge weights, provided our algorithm is given access to an oracle for the decision problem. Our result is built on the work of [AV18], which used planarity of the input graph critically; in fact, in three different ways. The main challenge was to adapt these steps to general graphs by appropriately trading planarity with the use of the decision oracle.
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