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 most relevant works are by van Leeuwen and Schoone @cite_6 , and Oda and Watanabe @cite_18 for Hamiltonian cycles, and by Bonnet and Miltzow @cite_0 for matchings. They proved, with elegant arguments, the following results.
@cite_0 @cite_18 @cite_6
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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. We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of G(n,p) in order to typically find a subgraph possessing a given target property. We show that if p≥lnn+lnlnn+ω(1)n, then one can find a Hamilton cycle with high probability after exposing (1+o(1))n edges. Our result is tight in both p and the number of exposed edges. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016 We describe a fixed-point based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley, the Mendelsohn-Dulmage theorem, the Kundu-Lawler theorem, Tarski's fixed-point theorem, the Cantor-Bernstein theorem, Pym's linking theorem, or the monochromatic path theorem of In this framework, we formulate a matroid-generalization of the stable marriage theorem and study the lattice structure of generalized stable matchings. Based on the theory of lattice polyhedra and blocking polyhedra, we extend results of Vande Vate and Rothblum on the bipartite stable matching polytope. Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian cycle instances with the Gn,m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the Gn,m phase transition which asymptotically produces a high frequency of hard instances.