Balanced bisubmodular systems and bidirected flows

For a nonempty finite set V let 3(v) be the set of all the ordered pairs of disjoint subsets of V, i.e., 3(v) = {(X,Y)\X, Y subset of or equal to V,X boolean AND Y = 0}. We define two operations, reduced union (sic) and intersection (sic), on 3(v) as follows: for each (X-i,Y-i) is an element of 3(v) (i = 1, 2) (X-1,Y-1) box with top missing (X-2,Y-2) = ((X-1 boolean OR X-2) - (Y-1 boolean OR Y-2), (Y-1 boolean OR Y-2) - (X-1 boolean OR X-2)), (X-1, Y-1) box with bottom missing (X-2, Y-2) = (X-1 boolean AND X-2, Y-1 boolean AND Y-2). Also, for a {box with top missing, box with bottom missing}-closed family F subset of or equal to 3(v) a function f : F --> R is called bisubmodular if for each (X-i,Y-i) is an element of F (i = 1,2) we have F(X-1,Y-1) + f(X-2,Y-2) greater than or equal to f((X-1,Y-1) box with top missing (X-2,Y-2)) + f((X-1,Y-1) box with bottom missing (X-2,Y-2)). For a {box with top missing, box with bottom missing)-closed family F subset of or equal to 3(v) with (0,0) is an element of F and a so-called bisubmodular function f : F --> R on F with f(0, 0) = 0, the pair (F, f) is called a bisubmodular system on V. In this paper we consider two classes of bisubmodular systems which are closely related to base polyhedra. The first one is the class of balanced bisubmodular system. We give a characterization of balanced bisubmodular systems and show that their associated polyhedra are the convex hulls of reflections of base polyhedra. The second one is that of cut functions of bidirected networks. It is shown that the polyhedron determined by the cut function of a bidirected network is the set of the boundaries of hows in the bidirected network and is a projection of a section of a base polyhedron of boundaries of an associated ordinary network.