SIAM J. DISCRETE MATH. c  2006 Society for Industrial and Applied Mathematics Vol. 19, No. 4, pp. 1065–1073 BISUBMODULAR FUNCTION MINIMIZATION ∗ SATORU FUJISHIGE † AND SATORU IWATA ‡ Abstract. This paper presents the first combinatorial polynomial algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. Since the rank functions of delta-matroids are bisubmodular, the scaling algorithm naturally leads to the first combinatorial polynomial algorithm for testing membership in delta-matroid polyhedra. Key words. bisubmodular function, delta-matroid, scaling algorithm AMS subject classification. 90C27 DOI. 10.1137/S0895480103426339 1. Introduction. Let V be a finite nonempty set of cardinality n and 3 V denote the set of ordered pairs of disjoint subsets of V . Two binary operations  and  on 3 V are defined by (X 1 ,Y 1 )  (X 2 ,Y 2 ) = ((X 1 ∪ X 2 )\(Y 1 ∪ Y 2 ), (Y 1 ∪ Y 2 )\(X 1 ∪ X 2 )), (X 1 ,Y 1 )  (X 2 ,Y 2 )=(X 1 ∩ X 2 ,Y 1 ∩ Y 2 ). A function f :3 V → R is called bisubmodular if it satisfies f (X 1 ,Y 1 )+ f (X 2 ,Y 2 ) ≥ f ((X 1 ,Y 1 )  (X 2 ,Y 2 )) + f ((X 1 ,Y 1 )  (X 2 ,Y 2 )) for any (X 1 ,Y 1 ) and (X 2 ,Y 2 ) in 3 V . This paper presents the first combinatorial polynomial algorithm for minimizing bisubmodular functions. Examples of bisubmodular functions include the rank functions of delta-matroids introduced independently by Bouchet [3] and Chandrasekaran–Kabadi [6]. A delta- matroid is a set system (V, F ) with F being a nonempty family of subsets of V that satisfies the following exchange property: ∀F 1 ,F 2 ∈F , ∀v ∈ F 1 F 2 , ∃u ∈ F 1 F 2 : F 1 {u, v}∈F , where  denotes the symmetric difference. A slightly restricted set system with an additional condition ∅∈F had been introduced by Dress–Havel [11]. A member of F is called a feasible set of the delta-matroid. Note that the base and the independent- set families of a matroid satisfy this exchange property. Thus, a delta-matroid is a generalization of a matroid. Chandrasekaran–Kabadi [6] showed that the rank function  :3 V → Z defined by (X, Y ) = max{|X ∩ F |−|Y ∩ F || F ∈ F} ∗ Received by the editors April 25, 2003; accepted for publication (in revised form) September 22, 2005; published electronically February 3, 2006. A preliminary version of this paper appeared in Proceedings of the Eighth Conference on Integer Programming and Combinatorial Optimization, LNCS 2081, Springer-Verlag, Berlin, 2001, pp. 160–169. http://www.siam.org/journals/sidma/19-4/42633.html † Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan (fujishig @kurims.kyoto-u.ac.jp). ‡ Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan (iwata@ mist.i.u-tokyo.ac.jp). 1065