Institution: | a Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan b Division of Mathematical Sciences for Social Systems, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan c Department of Medical Technology, Kochi Gakuen College, 292 Asahi-Tenjincho, Kochi 780-0955, Japan d Department of Systems Science, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan |
Abstract: | It has widely been recognized that submodular set functions and base polyhedra associated with them play fundamental and important roles in combinatorial optimization problems. In the present paper, we introduce a generalized concept of base polyhedron. We consider a class of pointed convex polyhedra in RV whose edge vectors have supports of size at most 2. We call such a convex polyhedron a polybasic polyhedron. The class of polybasic polyhedra includes ordinary base polyhedra, submodular/supermodular polyhedra, generalized polymatroids, bisubmodular polyhedra, polybasic zonotopes, boundary polyhedra of flows in generalized networks, etc. We show that for a pointed polyhedron PRV, the following three statements are equivalent: - (1) P is a polybasic polyhedron.
- (2) Each face of P with a normal vector of the full support V is obtained from a base polyhedron by a reflection and scalings along axes.
- (3) The support function of P is a submodular function on each orthant of RV.
This reveals the geometric structure of polybasic polyhedra and its relation to submodularity. |