1,455 research outputs found

    Bounds on the maximum multiplicity of some common geometric graphs

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    We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

    An ETH-Tight Exact Algorithm for Euclidean TSP

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    We study exact algorithms for {\sc Euclidean TSP} in Rd\mathbb{R}^d. In the early 1990s algorithms with nO(n)n^{O(\sqrt{n})} running time were presented for the planar case, and some years later an algorithm with nO(n11/d)n^{O(n^{1-1/d})} running time was presented for any d2d\geq 2. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on {\sc Euclidean TSP}, except for a lower bound stating that the problem admits no 2O(n11/dϵ)2^{O(n^{1-1/d-\epsilon})} algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of {\sc Euclidean TSP} by giving a 2O(n11/d)2^{O(n^{1-1/d})} algorithm and by showing that a 2o(n11/d)2^{o(n^{1-1/d})} algorithm does not exist unless ETH fails.Comment: To appear in FOCS 201

    Stable marriage and roommates problems with restricted edges: complexity and approximability

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    In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs. Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs

    Dimers, Tilings and Trees

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    Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure

    Ising Model Observables and Non-Backtracking Walks

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    This paper presents an alternative proof of the connection between the partition function of the Ising model on a finite graph GG and the set of non-backtracking walks on GG. The techniques used also give formulas for spin-spin correlation functions in terms of non-backtracking walks. The main tools used are Viennot's theory of heaps of pieces and turning numbers on surfaces.Comment: 33 pages, 11 figures. Typos and errors corrected, exposition improved, results unchange

    Quadri-tilings of the plane

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    We introduce {\em quadri-tilings} and show that they are in bijection with dimer models on a {\em family} of graphs {R}\{R^*\} arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called {\em triangular quadri-tilings}, as an interface model in dimension 2+2. Assigning "critical" weights to edges of RR^*, we prove an explicit expression, only depending on the local geometry of the graph RR^*, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of \cite{Kenyon1}. We also show that when edges of RR^* are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.Comment: Revised version, minor changes. 30 pages, 13 figure
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