We study an old geometric optimization problem in the plane. Given a perfect matching @math on a set of @math points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from @math and adds two non-crossing edges. Let @math and @math denote the minimum and maximum lengths of a flip sequence on @math , respectively. It has been proved by Bonnet and Miltzow (2016) that @math and by van Leeuwen and Schoone (1980) that @math . We prove that @math where @math is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound if the point set has sublinear spread. For a matching @math on @math points in convex position we prove that @math and @math ; these bounds are tight. Any bound on @math carries over to the bichromatic setting, while this is not necessarily true for @math . Let @math be a bichromatic matching. The best known upper bound for @math is the same as for @math , which is essentially @math . We prove that @math for points in convex position, and @math for semi-collinear points. The flip operation can also be defined on spanning trees. For a spanning tree @math on a convex point set we show that @math .
The @math upper bound of Theorem carries over to perfect matchings. As for lower bounds, Bonnet and Miltzow @cite_0 presented two matchings @math and @math such that @math and @math . The bound @math holds even if @math is a bichromatic matching, while the proof of @math does not generalize for the bichromatic setting.
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Friedland's Lower Matching Conjecture asserts that if @math is a @math --regular bipartite graph on @math vertices, and @math denotes the number of matchings of size @math , then @math where @math . When @math , this conjecture reduces to a theorem of Schrijver which says that a @math --regular bipartite graph on @math vertices has at least @math perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that @math In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra @math factor compared to the conjecture if @math is separated away from @math and @math , and is tight up to a constant factor if @math is separated away from @math . We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to @math --biregular bipartite graphs.