176 research outputs found

    Excluding Surfaces as Minors in Graphs

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    We introduce an annotated extension of treewidth that measures the contribution of a vertex set XX to the treewidth of a graph G.G. This notion provides a graph distance measure to some graph property P\mathcal{P}: A vertex set XX is a kk-treewidth modulator of GG to P\mathcal{P} if the treewidth of XX in GG is at most kk and its removal gives a graph in P.\mathcal{P}.This notion allows for a version of the Graph Minors Structure Theorem (GMST) that has no need for apices and vortices: KkK_k-minor free graphs are those that admit tree-decompositions whose torsos have ckc_{k}-treewidth modulators to some surface of Euler-genus ck.c_{k}. This reveals that minor-exclusion is essentially tree-decomposability to a ``modulator-target scheme'' where the modulator is measured by its treewidth and the target is surface embeddability. We then fix the target condition by demanding that Σ\Sigma is some particular surface and define a ``surface extension'' of treewidth, where \Sigma\mbox{-}\mathsf{tw}(G) is the minimum kk for which GG admits a tree-decomposition whose torsos have a kk-treewidth modulator to being embeddable in Σ.\Sigma.We identify a finite collection DΣ\mathfrak{D}_{\Sigma} of parametric graphs and prove that the minor-exclusion of the graphs in DΣ\mathfrak{D}_{\Sigma} precisely determines the asymptotic behavior of {\Sigma}\mbox{-}\mathsf{tw}, for every surface Σ.\Sigma. It follows that the collection DΣ\mathfrak{D}_{\Sigma} bijectively corresponds to the ``surface obstructions'' for Σ,\Sigma, i.e., surfaces that are minimally non-contained in $\Sigma.

    Chromatic numbers of Cayley graphs of abelian groups: A matrix method

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    In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of Heuberger and others, in such cases the graph can be represented by an m×rm\times r integer matrix, where we call mm the dimension and rr the rank. Adding or subtracting rows produces a graph homomorphism to a graph with a matrix of smaller dimension, thereby giving an upper bound on the chromatic number of the original graph. In this article we develop the foundations of this method. In a series of follow-up articles using this method, we completely determine the chromatic number in cases with small dimension and rank; prove a generalization of Zhu's theorem on the chromatic number of 66-valent integer distance graphs; and provide an alternate proof of Payan's theorem that a cube-like graph cannot have chromatic number 3.Comment: 17 page

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Fully dynamic approximation schemes on planar and apex-minor-free graphs

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    The classic technique of Baker [J. ACM '94] is the most fundamental approach for designing approximation schemes on planar, or more generally topologically-constrained graphs, and it has been applied in a myriad of different variants and settings throughout the last 30 years. In this work we propose a dynamic variant of Baker's technique, where instead of finding an approximate solution in a given static graph, the task is to design a data structure for maintaining an approximate solution in a fully dynamic graph, that is, a graph that is changing over time by edge deletions and edge insertions. Specifically, we address the two most basic problems -- Maximum Weight Independent Set and Minimum Weight Dominating Set -- and we prove the following: for a fully dynamic nn-vertex planar graph GG, one can: * maintain a (1ε)(1-\varepsilon)-approximation of the maximum weight of an independent set in GG with amortized update time f(ε)no(1)f(\varepsilon)\cdot n^{o(1)}; and, * under the additional assumption that the maximum degree of the graph is bounded at all times by a constant, also maintain a (1+ε)(1+\varepsilon)-approximation of the minimum weight of a dominating set in GG with amortized update time f(ε)no(1)f(\varepsilon)\cdot n^{o(1)}. In both cases, f(ε)f(\varepsilon) is doubly-exponential in poly(1/ε)\mathrm{poly}(1/\varepsilon) and the data structure can be initialized in time f(ε)n1+o(1)f(\varepsilon)\cdot n^{1+o(1)}. All our results in fact hold in the larger generality of any graph class that excludes a fixed apex-graph as a minor.Comment: 37 pages, accepted to SODA '2

    Dense RGB-D SLAM and object localisation for robotics and industrial applications

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    Dense reconstruction and object localisation are two critical steps in robotic and industrial applications. The former entails a joint estimation of camera egomotion and the structure of the surrounding environment, also known as Simultaneous Localisation and Mapping (SLAM), and the latter aims to locate the object in the reconstructed scenes. This thesis addresses the challenges of dense SLAM with RGB-D cameras and object localisation towards robotic and industrial applications. Camera drift is an essential issue in camera egomotion estimation. Due to the accumulated error in camera pose estimation, the estimated camera trajectory is inaccurate, and the reconstruction of the environment is inconsistent. This thesis analyses camera drift in SLAM under the probabilistic inference framework and proposes an online map fusion strategy with standard deviation estimation based on frame-to-model camera tracking. The camera pose is estimated by aligning the input image with the global map model, and the global map merges the information in the images by weighted fusion with standard deviation modelling. In addition, a pre-screening step is applied before map fusion to preclude the adverse effect of accumulated errors and noises on camera egomotion estimation. Experimental results indicated that the proposed method mitigates camera drift and improves the global consistency of camera trajectories. Another critical challenge for dense RGB-D SLAM in industrial scenarios is to handle mechanical and plastic components that usually have reflective and shiny surfaces. Photometric alignment in frame-to-model camera tracking tends to fail on such objects due to the inconsistency in intensity patterns of the images and the global map model. This thesis addresses this problem and proposes RSO-SLAM, namely a SLAM approach to reflective and shiny object reconstruction. RSO-SLAM adopts frame-to-model camera tracking and combines local photometric alignment and global geometric registration. This study revealed the effectiveness and excellent performance of the proposed RSO-SLAM on both plastic and metallic objects. In addition, a case study involving the cover of a electric vehicle battery with metallic surface demonstrated the superior performance of the RSO-SLAM approach in the reconstruction of a common industrial product. With the reconstructed point cloud model of the object, the problem of object localisation is tackled as point cloud registration in the thesis. Iterative Closest Point (ICP) is arguably the best-known method for point cloud registration, but it is susceptible to sub-optimal convergence due to the multimodal solution space. This thesis proposes the Bees Algorithm (BA) enhanced with the Singular Value Decomposition (SVD) procedure for point cloud registration. SVD accelerates the speed of the local search of the BA, helping the algorithm to rapidly identify the local optima. It also enhances the precision of the obtained solutions. At the same time, the global outlook of the BA ensures adequate exploration of the whole solution space. Experimental results demonstrated the remarkable performance of the SVD-enhanced BA in terms of consistency and precision. Additional tests on noisy datasets demonstrated the robustness of the proposed procedure to imprecision in the models

    On the density of matroids omitting a complete-graphic minor

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    We show that, if MM is a simple rank-nn matroid with no \ell-point line minor and no minor isomorphic to the cycle matroid of a tt-vertex complete graph, then the ratio M/n|M| / n is bounded above by a singly exponential function of \ell and tt. We also bound this ratio in the special case where MM is a frame matroid, obtaining an answer that is within a factor of two of best-possible.Comment: 25 page

    Variants of the Square Peg Problem

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    Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Juan Carlos Naranjo del Val[en] The Square Peg Problem, also known as Toeplitz’ Conjecture, is an unsolved problem in the mathematical areas of geometry and topology that states the following: Every simple closed curve in the plane inscribed a square. Even though it seems like an innocent statement, it requires a lot of technical knowledge to proof even when applying certain smoothness conditions to the curve. Over time, variants of this problem have emerged. Some of them offer very interesting results with beautiful proofs. We intend on giving a general historical overview about the Square Peg Problem and the most known variants. Then we will explore the variants related to the inscription of rectangles and triangles and show a few strong results

    Approximating branchwidth on parametric extensions of planarity

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    The \textsl{branchwidth} of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph classes, further than planar graphs as follows: Let H0H_{0} be a graph embeddedable in the projective plane and H1H_{1} be a graph embeddedable in the torus. We prove that every {H0,H1}\{H_{0},H_{1}\}-minor free graph GG contains a subgraph GG' where the difference between the branchwidth of GG and the branchwidth of GG' is bounded by some constant, depending only on H0H_{0} and H1H_{1}. Moreover, the graph GG' admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving an EPTAS for branchwidth: For {H0,H1}\{H_{0},H_{1}\}-minor free graphs, there is a function f ⁣:NNf\colon\mathbb{N}\to\mathbb{N} and a (1+ϵ)(1+\epsilon)-approximation algorithm for branchwidth, running in time O(n3+f(1ϵ)n),\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n), for every ϵ>0\epsilon>0

    A computational multi-scale approach for brittle materials

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    Materials of industrial interest often show a complex microstructure which directly influences their macroscopic material behavior. For simulations on the component scale, multi-scale methods may exploit this microstructural information. This work is devoted to a multi-scale approach for brittle materials. Based on a homogenization result for free discontinuity problems, we present FFT-based methods to compute the effective crack energy of heterogeneous materials with complex microstructures
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