Even factors, jump systems, and discrete convexity
Journal of Combinatorial Theory Series B
Neighbor Systems and the Greedy Algorithm
SIAM Journal on Discrete Mathematics
An algorithm for (n-3)-connectivity augmentation problem: Jump system approach
Journal of Combinatorial Theory Series B
A proof of Cunningham's conjecture on restricted subgraphs and jump systems
Journal of Combinatorial Theory Series B
A simple algorithm for finding a maximum triangle-free 2-matching in subcubic graphs
Discrete Optimization
Neighbor Systems, Jump Systems, and Bisubmodular Polyhedra
SIAM Journal on Discrete Mathematics
Matching problems with delta-matroid constraints
CATS '12 Proceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 128
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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.