Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

Bisubmodular functions are a natural “directed”, or “signed”, extension of submodular functions with several applications. Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM). However, they were able to extend only the weakly polynomial version of IFF to BSFM. Here we investigate the difficulty that prevented them from also extending the strongly polynomial version of IFF to BSFM, and we show a way around the difficulty. This new method gives a somewhat simpler strongly polynomial SFM algorithm, as well as the first combinatorial strongly polynomial algorithm for BSFM. This further leads to extending Iwata’s fully combinatorial version of IFF to BSFM. The research of S. T. McCormick was supported by an NSERC Operating Grant. The research of S. Fujishige was supported by a Grant-in-Aid of the Ministry of Education, Culture, Science and Technology of Japan.