14,874 research outputs found

    Complexity of metric dimension on planar graphs

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    © . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The metric dimension of a graph G is the size of a smallest subset L ¿ V (G) such that for any x, y ¿ V (G) with x =/ y there is a z ¿ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, prior to our work the computational complexity of determining the metric dimension of a graph was still very unclear. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on planar graphs of maximum degree 6 is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.Peer ReviewedPostprint (author's final draft

    Weak hyperbolicity of cube complexes and quasi-arboreal groups

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    We examine a graph Γ\Gamma encoding the intersection of hyperplane carriers in a CAT(0) cube complex X~\widetilde X. The main result is that Γ\Gamma is quasi-isometric to a tree. This implies that a group GG acting properly and cocompactly on X~\widetilde X is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove asymptotic finite-dimensionality for finitely-generated groups acting on finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded asymptotic dimension. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of Γ\Gamma.Comment: Corrections in Sections 2 and 4. Simplification in Section

    Metric Dimension for Gabriel Unit Disk Graphs is NP-Complete

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    We show that finding a minimal number of landmark nodes for a unique virtual addressing by hop-distances in wireless ad-hoc sensor networks is NP-complete even if the networks are unit disk graphs that contain only Gabriel edges. This problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The Gabriel edges of a unit disc graph induce a planar O(\sqrt{n}) distance and an optimal energy spanner. This is one of the most interesting restrictions of Metric Dimension in the context of wireless multi-hop networks.Comment: A brief announcement of this result has been published in the proceedings of ALGOSENSORS 201

    Geometric versions of the 3-dimensional assignment problem under general norms

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    We discuss the computational complexity of special cases of the 3-dimensional (axial) assignment problem where the elements are points in a Cartesian space and where the cost coefficients are the perimeters of the corresponding triangles measured according to a certain norm. (All our results also carry over to the corresponding special cases of the 3-dimensional matching problem.) The minimization version is NP-hard for every norm, even if the underlying Cartesian space is 2-dimensional. The maximization version is polynomially solvable, if the dimension of the Cartesian space is fixed and if the considered norm has a polyhedral unit ball. If the dimension of the Cartesian space is part of the input, the maximization version is NP-hard for every LpL_p norm; in particular the problem is NP-hard for the Manhattan norm L1L_1 and the Maximum norm LL_{\infty} which both have polyhedral unit balls.Comment: 21 pages, 9 figure

    Localization game on geometric and planar graphs

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    The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph GG we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant ζ(G)\zeta (G), defined as the least number of cops needed to localize the robber on a graph GG, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth 22 and unbounded ζ(G)\zeta (G). On a positive side, we prove that ζ(G)\zeta (G) is bounded by the pathwidth of GG. We then show that the algorithmic problem of determining ζ(G)\zeta (G) is NP-hard in graphs with diameter at most 22. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane

    Travelling on Graphs with Small Highway Dimension

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    We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]

    Quantum automorphism groups of small metric spaces

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    To any finite metric space XX we associate the universal Hopf \c^*-algebra HH coacting on XX. We prove that spaces XX having at most 7 points fall into one of the following classes: (1) the coaction of HH is not transitive; (2) HH is the algebra of functions on the automorphism group of XX; (3) XX is a simplex and HH corresponds to a Temperley-Lieb algebra; (4) XX is a product of simplexes and HH corresponds to a Fuss-Catalan algebra.Comment: 22 page
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