Discrete convexity in joint winner property

Abstract

In this paper, we reveal a relation between joint winner property (JWP) in the field of valued constraint satisfaction problems (VCSPs) and M ♮ -convexity in the field of discrete convex analysis (DCA). We introduce the M ♮ -convex completion problem, and show that a function f satisfying the JWP is Z-free if and only if a certain function f associated with f is M ♮ -convex completable. This means that if a function is Z-free, then the function can be minimized in polynomial time via M ♮ -convex intersection algorithms. Furthermore we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values.• To describe the connection of JWP and M ♮ -convexity, we introduce the M ♮ -convex completion problem, and give a characterization of M ♮ -convex completability.• By utilizing a DCA interpretation of JWP, we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values.This study will hopefully be the first step towards fruitful interactions between VCSPs and DCA.Notations. Let R and R + denote the sets of reals and nonnegative reals, respectively. In this paper, functions can take the infinite value +∞, where a < +∞, a + ∞ = +∞ for a ∈ R, and 0 · (+∞) = 0. Let R := R ∪ {+∞} and R +

The quadratic M-convexity testing problem

“…A binary VCSP function F satisfying the joint winner property can be transformed to a function represented as the sum two special M-convex functions of the form (1) with an infinite quadratic coefficient. This fact explains the polynomial-time solvability of F (see [6] for details). The class of functions represented as the sum of two general quadratic M-convex functions corresponds to a new tractable class of binary VCSPs.…”

“…Quadratic Mconvex functions have a close relationship with tree metrics [5], which is an important concept for mathematical analysis in phylogenetics (see e.g., [12]). Recently, Iwamasa-Murota-Živný [6] have revealed hidden quadratic M-convexity in valued constraint satisfaction problems (VCSPs) with joint winner property [2], and presented a perspective to their polynomial-time solvability from discrete convex analysis.…”

A tractable class of binary VCSPs via M-convex intersection

“…Thus, if F satisfies the JWP, then F can be minimized in polynomial time. Furthermore, Iwamasa, Murota, and Živný [20] revealed that Z-free instances are quadratic M 2 -representable.…”

“…Various kinds of submodularity induce tractable classes of language-based VCSP instances [22], and a larger class of such submodularity can be understood as L-convexity on certain graph structures [14]; see also [15]. On the other hand, Iwamasa, Murota, and Živný [20] have pointed out that M-convexity plays a role in hybrid VCSPs. They revealed the reason for the tractability of a VCSP instance satisfying the JWP from a viewpoint of M-convexity.…”

“…It is shown in [20] that a binary VCSP instance satisfying the JWP can be transformed to a quadratic M 2 -representable instance, † and two M-convex summands can be obtained in polynomial time. Here the following natural question arises: What binary VCSP instances are quadratic M 2 -representable?…”