A Min-Max Theorem for Bisubmodular Polyhedra

For a family ${\cal F} \subseteq 3^E$ closed with respect to the reduced union and intersection and for a bisubmodular function $f: {\cal F}\rightarrow \mbox{\bf R}$ with $(\emptyset,\emptyset) \in \cal F$ and $f(\emptyset,\emptyset)=0$, the bisubmodular polyhedron associated with $({\cal F},f)$ is given by \[ {\rm P}_*(f)=\{x\,|\,x\in\mbox{\bf R}^E\quad \forall (X,Y)\in {\cal F}: x(X,Y)\le f(X,Y)\}, \] where $x(X,Y)=\sum_{e\in X}x(e)-\sum_{e\in Y}x(e)$. We show a min--max relation that characterizes the distance between ${\rm P}_*(f)$ and a given point $x^0$ with respect to the $l_1$ norm: for any vector $x^0\in\mbox{\bf R}^E$, \[ \left. \min\left\{\sum_{e\in E}|x(e)-x^0(e)|\, \right| \,x\in{\rm P}_*(f)\right\} =\max\{x^0(X,Y)-f(X,Y)\,|\,(X,Y)\in{\cal F}\}, \] where if $f$ is integer valued and $x^0$ is integral, then the minimum is attained by an integral $x\in{\rm P}_*(f)$. This is in a sense equivalent to but is in a nicer symmetric form than a min--max theorem of Cunningham and Green-Krotki [Combinatorica, 11 (1991), pp. 219--230] shown to be associated with $b$-matching degree-sequence polyhedra and generalizes the well-known min--max theorem concerning a vector reduction of polymatroids and submodular systems. We also give an application of the theorem to a separable convex optimization problem on bisubmodular polyhedra.