Constructing a perfect matching is in random NC

Abstract

We show that the problem of constructing a perfect matching in a graph is in the complexity class Random NC; i.e., the problem is solvable in polylog time by a randomized parallel algorithm using a polynomial-bounded number of processors. We also show that several related problems lie in Random NC. These include:

  1. (i)

    Constructing a perfect matching of maximum weight in a graph whose edge weights are given in unary notation;

  2. (ii)

    Constructing a maximum-cardinality matching;

  3. (iii)

    Constructing a matching covering a set of vertices of maximum weight in a graph whose vertex weights are given in binary;

  4. (iv)

    Constructing a maximums-t flow in a directed graph whose edge weights are given in unary.

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Research supported by NSF Grant # DCR-8411954.

Research supported by a Weizmann Post-Doctoral fellowship, and by DARPA Grant N00039-83-C-1036.

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Karp, R.M., Upfal, E. & Wigderson, A. Constructing a perfect matching is in random NC. Combinatorica 6, 35–48 (1986). https://doi.org/10.1007/BF02579407

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