Neighbor systems, jump systems, and bisubmodular polyhedra

The concept of neighbor system, introduced by Hartvigsen in 2010, 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 first 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. It is shown that the problem can be solved in weakly polynomial time for a class of neighbor systems.