2,149 research outputs found

    A combinatorial approach to knot recognition

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    This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in a compact, accessible manner, and to show how to use it for computational purposes. In particular, we address how to determine colorability of a knot, and propose to use SAT solving to search for colorings. The computational complexity of the problem, both in theory and in our implementation, is discussed. In the last part, we explain how coloring can be utilized in knot recognition

    Recognising Multidimensional Euclidean Preferences

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    Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as multidimensional unfolding, has applications in economics, psychometrics, marketing, and many other fields. We study the problem of deciding whether a given preference profile is d-Euclidean. For the one-dimensional case, polynomial-time algorithms are known. We show that, in contrast, for every other fixed dimension d > 1, the recognition problem is equivalent to the existential theory of the reals (ETR), and so in particular NP-hard. We further show that some Euclidean preference profiles require exponentially many bits in order to specify any Euclidean embedding, and prove that the domain of d-Euclidean preferences does not admit a finite forbidden minor characterisation for any d > 1. We also study dichotomous preferencesand the behaviour of other metrics, and survey a variety of related work.Comment: 17 page

    On the pseudolinear crossing number

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    A drawing of a graph is {\em pseudolinear} if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The {\em pseudolinear crossing number} of a graph GG is the minimum number of pairwise crossings of edges in a pseudolinear drawing of GG. We establish several facts on the pseudolinear crossing number, including its computational complexity and its relationship to the usual crossing number and to the rectilinear crossing number. This investigation was motivated by open questions and issues raised by Marcus Schaefer in his comprehensive survey of the many variants of the crossing number of a graph.Comment: 12 page

    Exact Distance Oracles for Planar Graphs

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    We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation S[nlglgn,n2]S\in[n\lg\lg n,n^2], we show how to construct in O~(S)\tilde O(S) time a data structure of size O(S)O(S) that answers distance queries in O~(n/S)\tilde O(n/\sqrt S) time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever k[n,n)k\in[\sqrt n,n). * We provide a linear-space exact distance oracle for planar graphs with query time O(n1/2+eps)O(n^{1/2+eps}) for any constant eps>0. This is the first such data structure with provable sublinear query time. * For edge lengths at least one, we provide an exact distance oracle of space O~(n)\tilde O(n) such that for any pair of nodes at distance D the query time is O~(minD,n)\tilde O(min {D,\sqrt n}). Comparable query performance had been observed experimentally but has never been explained theoretically. Our data structures are based on the following new tool: given a non-self-crossing cycle C with c=O(n)c = O(\sqrt n) nodes, we can preprocess G in O~(n)\tilde O(n) time to produce a data structure of size O(nlglgc)O(n \lg\lg c) that can answer the following queries in O~(c)\tilde O(c) time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and extends a related data structure of Klein (SODA'05), which reports distances to the boundary of a face, rather than a cycle. The best distance oracles for planar graphs until the current work are due to Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For σ(1,4/3)\sigma\in(1,4/3) and space S=nσS=n^\sigma, we essentially improve the query time from n2/Sn^2/S to n2/S\sqrt{n^2/S}.Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, SODA 201

    The Complexity of Simultaneous Geometric Graph Embedding

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    Given a collection of planar graphs G1,,GkG_1,\dots,G_k on the same set VV of nn vertices, the simultaneous geometric embedding (with mapping) problem, or simply kk-SGE, is to find a set PP of nn points in the plane and a bijection ϕ:VP\phi: V \to P such that the induced straight-line drawings of G1,,GkG_1,\dots,G_k under ϕ\phi are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for R\exists\mathbb{R}, the existential theory of the reals. Hence the problem kk-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to kk-SGE, with the property that both numbers kk and nn are linear in the number of pseudolines. This implies not only the R\exists\mathbb{R}-hardness result, but also a 22Ω(n)2^{2^{\Omega (n)}} lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only 22Ω(n)2^{2^{\Omega (\sqrt{n})}}

    Toward the Rectilinear Crossing Number of KnK_n: New Drawings, Upper Bounds, and Asymptotics

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    Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersect in a point unless that point is an endpoint of all three. The rectilinear crossing number of K_n is the fewest number of edge crossings attainable over all rectilinear drawings of K_n. For each n we construct a rectilinear drawing of K_n that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of drawings of K_n with good asymptotics. Finally, we mention some old and new open problems.Comment: 13 Page
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