M-Convex Function on Generalized Polymatroid

Abstract

The concept of M-convex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of Dress-Wenzel (1990). In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems. The restriction of a function to {x ∈ Z V | x(V ) = k} for k ∈ Z is called a layer. We prove the M-convexity of each layer, and reveal that the minimizers in consecutive layers are closely related. Exploiting these properties, we can solve the optimization on layers efficiently.A number of equivalent exchange axioms are given for M-convex function on generalized polymatroid.

Identifying combinatorial valuations from aggregate demand

“…11 See Theorems 6.19, 6.42 and 11.5 of Murota (2003), originally proven by Murota and Shioura (2001), Murota (1996), Danilov, Koshevoy, and Lang (2003), Fujishige and Yang (2003), and Murota and Tamura (2003). V * g is convex, and hence has nonempty relative interior (i.e., nonempty interior relative to its affine hull aff V * g ), and any measurable set with nonempty interior in a Euclidean space has nonzero Lebesgue measure in that space, V * g has nonzero Lebesgue measure in aff V * g .…”

“…M -concavity (read "M -natural concavity") was introduced by Murota and Shioura (1999) and extensively studied by Murota (2003). M -concavity says that if bundle z contains more units of good j than does z , the sum of values of the bundles increases either when (i) a unit of j is transferred from bundle z to z , or (ii) a unit of good j is transferred from z to bundle z and, in exchange, a unit of some good j -of which z contains more units than z-is transferred from z to z.…”

Gross substitutability: An algorithmic survey

“…Murota and Shioura [35] define M -concave functions based on the concept of M -concavity of Murota [31]. They originally define M -concave functions on the integral lattice Z n , but for the purposes of this survey, we will consider their restriction to {0, 1} n :…”

Two-Sided Matching With Externalities: A Survey

“…These substitutability conditions have also been connected to the literature on discrete convexity. Fujishige and Yang [24] first connects gross substitutability of Kelso and Crawford [31] to M ♮ -convexity of Murota and Shioura [45], which is an equivalent variant of Murota [43,44]. Subsequent matching models that have used discrete convexity include (and are not confined to) Danilov et al [13], Fujishige and Tamura [23], Murota and Yokoi [46], and Kojima et al [37].…”