宮本 裕一郎

This paper discusses the problem of determining whether a given plane graph is a Delaunay graph, i.e., whether it is topologically equivalent to a Delaunay triangulation. There exist theorems which characterize Delaunay graphs and yield polynomial time algorithms for the problem only by solving some linear inequality systems. A polynomial time algorithm proposed by Hodgson, Rivin and Smith solves a linear inequality system given by Rivin, which is based on sophisticated arguments about hyperbolic geometry. Independently, Hiroshima, Miyamoto and Sugihara gave another linear inequality system and a polynomial time algorithm. Although their discussion is based on primitive arguments on Euclidean geometry, their proofs are long and intricate, unfortunately. In this paper, we give a simple proof of the theorem shown by Hiroshima et al. by employing the fixed point theorem.

Modularity proposed by Newman and Girvan is a quality function for community detection. Numerous heuristics for modularity maximization have been proposed because the problem is NP-hard. However, the accuracy of these heuristics has yet to be properly evaluated because computational experiments typically use large networks whose optimal modularity is unknown. In this study, we propose two powerful methods for computing a nontrivial upper bound of modularity. More precisely, our methods can obtain the optimal value of a linear programming relaxation of the standard integer linear programming for modularity maximization. The first method modifies the traditional row generation approach proposed by Grotschel and Wakabayashi to shorten the computation time. The second method is based on a row and column generation. In this method, we first solve a significantly small subproblem of the linear programming and iteratively add rows and columns. Owing to the speed and memory efficiency of these proposed methods, they are suitable for large networks. In particular, the second method consumes exceedingly small memory capacity, enabling us to compute the optimal value of the linear programming for the Power Grid network (consisting of 4941 vertices and 6594 edges) on a standard desktop computer.