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Register a new userAn Extension of the Birkhoff-von Neumann Theorem to Non-Bipartite Graphs
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Vijay V. Vazirani
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We prove that a fractional perfect matching in a non-bipartite graph can be written, in polynomial time, as a convex combination of perfect matchings. This extends the Birkhoff-von Neumann Theorem from bipartite to non-bipartite graphs. The algorithm of Birkhoff and von Neumann is greedy; it starts with the given fractional perfect matching and successively removes from it perfect matchings, with appropriate coefficients. This fails in non-bipartite graphs -- removing perfect matchings arbitrarily can lead to a graph that is non-empty but has no perfect matchings. Using odd cuts appropriately saves the day.
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We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
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