Knapsack Intersection Graphs and Efficient Computation of Their Maximal Cliques

Abstract

AbstractAs 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.KeywordsIntersection graphMax-tolerance graphKnapsack graphMaximal/maximum cliqueClique number