Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

Authors:
S. Thomas McCormick;Satoru Fujishige
Affiliations:
University of British Columbia, Sauder School of Business, V6T 1Z2, Vancouver, BC, Canada;Kyoto University, Research Institute for Mathematical Sciences, 606-8502, Kyoto, Japan
Venue:
Mathematical Programming: Series A and B
Year:
2009

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Abstract

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.