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.
@cite_0
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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. We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees. For any integers @math and @math , we show the existence of a set of @math vectors @math such that any embedding @math satisfying @math must have @math This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of @math of interest, since there is always an isometric embedding into dimension @math (either the identity map, or projection onto @math ). Previously such a lower bound was only known to hold against linear maps @math , and not for such a wide range of parameters @math [LN16]. The best previously known lower bound for general @math was @math [Wel74, Lev83, Alo03], which is suboptimal for any @math . A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least @math perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least @math perfect matchings.