5,243 research outputs found

    Colored Non-Crossing Euclidean Steiner Forest

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    Given a set of kk-colored points in the plane, we consider the problem of finding kk trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For k=1k=1, this is the well-known Euclidean Steiner tree problem. For general kk, a kρk\rho-approximation algorithm is known, where ρ1.21\rho \le 1.21 is the Steiner ratio. We present a PTAS for k=2k=2, a (5/3+ε)(5/3+\varepsilon)-approximation algorithm for k=3k=3, and two approximation algorithms for general~kk, with ratios O(nlogk)O(\sqrt n \log k) and k+εk+\varepsilon

    On the complexity of optimal homotopies

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    In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves γ1\gamma_1 and γ2\gamma_2 on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between γ1\gamma_1 and γ2\gamma_2 where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr\'echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting

    On the extremal properties of the average eccentricity

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    The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc(G)ecc (G) of a graph GG is the mean value of eccentricities of all vertices of GG. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease ecc(G)ecc (G). Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure

    On the Spectral Gap of a Quantum Graph

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    We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature
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