二元次模函数的表示能力
The Expressive Power of Binary Submodular Functions
摘要：
它以前一直是一个开放的问题，是否所有的布尔子模函数可以分解成一个总和的二元子模函数在一个可能更大的一组变量。这个问题已经在计算机科学的几个不同的背景下被考虑过，包括计算机视觉，人工智能和伪布尔优化。使用值约束的表达能力和函数的某些代数性质之间的联系，我们否定地回答了这个问题。我们的结果有几个推论。首先，我们精确地确定了4元的哪些子模函数可以用二元子模函数表示。接下来，我们确定了一个新的类的子模函数的任意arities，可以表示为二进制子模函数，因此有效地最小化使用所谓的可表达性减少的最小割问题。更重要的是，我们的研究结果意味着这种减少的限制，并建立第一次，它不能被用来在一般情况下，以尽量减少任意次模函数。最后，我们驳斥了Promislow和Young关于布尔次模函数锥的极端射线结构的一个猜想。
英文摘要：
It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular functions of arity 4 can be expressed by binary submodular functions. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish for the first time that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.
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