Bisubmodular Function Minimization
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Towards minimizing k-submodular functions
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
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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-Krótki [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.