Induction of M-convex functions by linking systems

  • Authors:

  • Yusuke Kobayashi;Kazuo Murota

  • Affiliations:

  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

Induction (or transformation) by bipartite graphs is one of the most important operations on matroids, and it is well known that the induction of a matroid by a bipartite graph is again a matroid. As an abstract form of this fact, the induction of a matroid by a linking system is known to be a matroid. M-convex functions are quantitative extensions of matroidal structures, and they are known as discrete convex functions. As with matroids, it is known that the induction of an M-convex function by networks generates an M-convex function. As an abstract form of this fact, this paper shows that the induction of an M-convex function by linking systems generates an M-convex function. Furthermore, we show that this result also holds for M-convex functions on constant-parity jump systems. Previously known operations such as aggregation, splitting, and induction by networks can be understood as special cases of this construction.