Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

• Published in: Mathematical programming Vol. 122; no. 1; pp. 87 - 120
• Main Authors: McCormick, S. Thomas, Fujishige, Satoru
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.03.2010; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Exact sciences and technology; Flows in networks. Combinatorial problems; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Optimization algorithms; Polynomials; Studies; Theoretical; Secondary: 90C27; 68W40; Primary: 65K05; Submodular function; Function minimization; Polynomial method; Combinatorial optimization; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-008-0242-9

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.