次模集合函数的单调分解
Monotonic Decompositions of Submodular Set Functions
AI总结：
本文将有限次模集合函数的单调分解结果推广到分析框架，引入无穷交替集合函数新类，证明其可分解为递增与递减次模函数之和或两个递增次模函数之差，并研究了无向图加权割函数的分解结构。
摘要：
次模集合函数无疑是组合优化中最重要的构建模块之一。有些令人惊讶的是，这类函数的连续对应物也出现在一条分析研究路线中，并在有限可加测度、非线性积分和电容理论中得到了应用。最近，这两个分支之间建立了许多联系，本文旨在将有限集合上次模集合函数的进一步结果推广到分析框架中。我们首先将拟阵的对偶性概念推广到次模集合函数，并刻画了一个次模集合函数唯一确定的分解，即分解为一个非负电荷与一个递增次模集合函数之和，其中电荷是极大的。然后，我们描述了无穷交替集合函数的基本性质，这是次模集合函数的一个子类，可作为覆盖函数的分析对应物。通过在定义中放宽单调性假设，我们引入了一类具有显著结构性质的新次模函数，其中包括图的加权割函数等。我们证明，与无穷域上的一般次模集合函数不同，任何无穷交替集合函数都可以写成一个递增次模函数与一个递减次模函数之和，或者写成两个递增次模函数之差，从而推广了有限情形下单调分解的结果。最后，受其与图参数（如割的最大规模和分数三角形填充的最大规模）之间联系的启发，我们研究了无向图加权割函数的此类分解的结构。
英文摘要：
Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they found applications in the theory of finitely additive measures, nonlinear integrals, and electric capacities. Recently, a number of connections between these two branches have been established, and the aim of this paper is to generalize further results on submodular set functions on finite sets to the analytic setting. We first extend the notion of duality of matroids to submodular set functions, and characterize the uniquely determined decomposition of a submodular set function into the sum of a nonnegaive charge and an increasing submodular set function in which the charge is maximal. Then, we describe basic properties of infinite-alternating set functions, a subclass of submodular set functions that serves as an analytic counterpart of coverage functions. By relaxing the monotonicity assumption in the definition, we introduce a new class of submodular functions with distinguished structural properties that includes, among others, weighted cut functions of graphs. We prove that, unlike general submodular set functions over an infinite domain, any infinite-alternating set function can be written as the sum of an increasing and a decreasing submodular function or as the difference of two increasing submodular functions, thus giving extension of results on monotonic decompositions in the finite case. Finally, motivated by its connections to graph parameters such as the maximum size of a cut and the maximum size of a fractional triangle packing, we study the structure of such decompositions for weighted cut functions of undirected graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
