一类易于处理的基于M-凸交的二元VCSP
A tractable class of binary VCSPs via M-convex intersection
摘要：
二进制VCSP是一个通用的框架，用于表示为一元和二元成本函数之和的函数的最小化问题。VCSP研究的一条重要路线是研究什么函数可以在多项式时间内求解。Cooper和Jumivnoun根据“三角形”的概念对二进制VCSP实例的易处理性进行了分类，并表明唯一有趣的易处理情况是由联合赢家属性（JWP）引起的情况。最近，Iwamasa，Murota，和Jakivnanwu将VCSP和离散凸分析联系起来，表明满足JWP的函数可以转换为表示为两个二次M-凸函数之和的函数，如果每个M-凸函数的值预言给定，则可以通过M-凸相交算法在多项式时间内最小化。在本文中，我们给出了一个算法的答案，一个自然的问题：什么样的二元有限值CSP的情况下，可以表示为两个二次M-凸函数的和，并可以解决在多项式时间通过一个M-凸相交算法？我们通过设计一个多项式时间算法来解决这个问题，该算法用于在可表示的情况下获得表示的具体形式。我们的结果提出了一个更大的听话类的二进制有限值CSP，它正确地包含JWP类。
英文摘要：
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 and 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, and 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 quadratic 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 finite-valued CSP instances can be represented as the sum of two quadratic M-convex functions and can be solved in polynomial time via an M-convex intersection algorithm? 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 finite-valued CSPs, which properly contains the JWP class.
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