线性规划的性质{A}与对偶
Property {A} and duality in linear programming
摘要：
性质A是群和度量空间的弱顺从性的一种形式，作为著名的诺维科夫高等签名猜想的一种方法而引入，诺维科夫高等签名猜想是拓扑学中最重要的未解决问题之一。我们证明了性质A可以归结为有限图上的一系列线性规划最优化问题。我们探讨的双重问题，这原来有有趣的解释组合问题的最大总供应流量的网络。利用等周不等式，将对偶问题与图的Cheeger常数联系起来，并探讨了图的对称性所起的作用，从而得到了扩张图与无性质A的图之间区别的一个显著特征.性质A是一种新的与图论相关的图的连通性度量。对偶线性问题可以使用多种方法来解决，我们在几个启发性的例子中证明了这一点。作为演示的权力，这种线性规划的方法，我们给优雅的证明，Nowak和Willett定理的图没有属性A。
英文摘要：
Property A is a form of weak amenability for groups and metric spaces introduced as an approach to the famous Novikov higher signature conjecture, one of the most important unsolved problems in topology. We show that property A can be reduced to a sequence of linear programming optimization problems on finite graphs. We explore the dual problems, which turn out to have interesting interpretations as combinatorial problems concerning the maximum total supply of flows on a network. Using isoperimetric inequalities, we relate the dual problems to the Cheeger constant of the graph and explore the role played by symmetry of a graph to obtain a striking characterization of the difference between an expander and a graph without property A. Property A turns out to be a new measure of connectivity of a graph that is relevant to graph theory. The dual linear problems can be solved using a variety of methods, which we demonstrate on several enlightening examples. As a demonstration of the power of this linear programming approach we give elegant proofs of theorems of Nowak and Willett about graphs without property A.
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