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Knapsack Intersection Graphs and Efficient Computation of Their Maximal Cliques

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Computational Modeling of Objects Presented in Images. Fundamentals, Methods, and Applications (CompIMAGE 2014)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8641))

Abstract

As an image can easily be modeled by its adjacency graph, graph theory and algorithms on graphs are widely used in imaging sciences. In this paper we define a knapsack graph, which is an intersection graph of integer translates of knapsack polygons, and consider the maximal clique problem on such graphs. A major application of intersection graphs is found in visualization of relations among objects in a scene. Efficient algorithms for the maximal clique problem are applicable to problems of computer graphics and image analysis, while properties of the knapsack polygon have been used in obtaining theoretical results in discrete geometry for computer imagery. We first show that the maximal clique problem on knapsack graphs is equivalent to the maximal clique problem on intersection graphs of homothetic right triangles. The latter was shown to be equivalent to the maximal clique problem on max-tolerance graphs and solvable in optimal O(n 3) time [28]. Thus, if the linear constraints defining the knapsack polygons are known, then the maximal clique problem on knapsack graphs can be solved using the algorithm from [28]. If the polygons are given by lists of their vertices and the defining constraints are unknown, we show how these can be found efficiently in computation time bounded by a low degree polynomial in the polygons size.

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Authors and Affiliations

  1. Mathematics Department, SUNY Buffalo State College, Buffalo, NY, 14222, USA

    Valentin E. Brimkov

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  1. Valentin E. Brimkov
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Editors and Affiliations

  1. Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, 15213, Pittsburgh, PA, USA

    Yongjie Jessica Zhang

  2. Faculdade de Engenharia, Departamento de Engenharia Mecânica, Universidade do Porto, Rua Dr. Roberto Frias, S/N, 4200-465, Porto, Portugal

    João Manuel R. S. Tavares

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Brimkov, V.E. (2014). Knapsack Intersection Graphs and Efficient Computation of Their Maximal Cliques. In: Zhang, Y.J., Tavares, J.M.R.S. (eds) Computational Modeling of Objects Presented in Images. Fundamentals, Methods, and Applications. CompIMAGE 2014. Lecture Notes in Computer Science, vol 8641. Springer, Cham. https://doi.org/10.1007/978-3-319-09994-1_16

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  • DOI: https://doi.org/10.1007/978-3-319-09994-1_16

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