Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions. An important line of VCSP research is to investigate what functions can be solved in polynomial time. Cooper–Živný classified the tractability of binary VCSP instances according to the concept of “triangle,” and showed that the only interesting tractable case is the one induced by the joint winner property (JWP). Recently, Iwamasa–Murota–Živný made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given. In this paper, we give an algorithmic answer to a natural question: What binary VCSP instances can be solved in polynomial time via an M-convex intersection algorithm? Under a natural condition, we solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case. Our result presents a larger tractable class of binary VCSPs, which properly contains the JWP class. We also show the co-NP-hardness of testing the representability of a given binary VCSP instance as the sum of two M-convex functions.