Neighbor Systems, Jump Systems, and Bisubmodular Polyhedra
The concept of neighbor system, introduced by Hartvigsen (2009), is a set of integral vectors satisfying a certain combinatorial property. In this paper, we reveal the relationship of neighbor systems with jump systems and with bisubmodular polyhedra. We firstly prove that for every neighbor system, there exists a jump system which has the same neighborhood structure as the original neighbor system. This shows that the concept of neighbor system is essentially equivalent to that of jump system. We next show that the convex closure of a neighbor system is an integral bisubmodular polyhedron. In addition, we give a characterization of neighbor systems using bisubmodular polyhedra. Finally, we consider the problem of minimizing a separable convex function on a neighbor system. By using the relationship between neighbor systems and jump systems shown in this paper, we prove that the problem can be solved in weakly-polynomial time for a class of neighbor systems.
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- 10.Edmonds, J.: Submodular functions, matroids, and certain polyhedra. Combinatorial Structures and their Applications, pp. 69–87. Gordon and Breach, New York (1970)Google Scholar
- 14.Hartvigsen, D.: Neighbor system and the greedy algorithm (extended abstract), RIMS Kôkyûroku Bessatsu (to appear, 2010)Google Scholar
- 16.Kobayashi, Y., Szabo, J., Takazawa, K.: A proof to Cunningham’s conjecture on restricted subgraphs and jump systems. TR-2010-04. Egervary Research Group, Budapest (2010)Google Scholar