## Abstract

The concept of a jump system, introduced by Bouchet and Cunningham [SIAM J. Discrete Math., 8 (1995), pp. 17-32], is a set of integer points with a certain exchange property. In this paper, we discuss several linear and convex optimization problems on jump systems and show that these problems can be solved in polynomial time under the assumption that a membership oracle for a jump system is available. We first present a polynomial-time implementation of the greedy algorithm for the minimization of a linear function. We then consider the minimization of a separable-convex function on a jump system and propose the first polynomial-time algorithm for this problem. The algorithm is based on the domain reduction approach developed in Shioura [Discrete Appl. Math., 84 (1998), pp. 215-220]. We finally consider the concept of M-convex functions on constant-parity jump systems which has been recently proposed by Murota [SIAM J. Discrete Math., 20 (2006), pp. 213-226]. It is shown that the minimization of an M-convex function can be solved in polynomial time by the domain reduction approach.

Original language | English |
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Pages (from-to) | 504-522 |

Number of pages | 19 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 |

Externally published | Yes |

## Keywords

- Bisubmodular function
- Bisubmodular polyhedron
- Discrete convex function
- Jump system
- Polynomial-time algorithm

## ASJC Scopus subject areas

- Mathematics(all)