P中问题的多项式规模线性规划
Polynomial size linear programs for problems in P
摘要：
无向图$G=(V,E)$中的完美匹配是来自$E$的顶点不相交边的集合，其包括$V$中的所有顶点。完美匹配问题是决定$G$是否具有这样的匹配。最近Rothvovich证明了Edmonds匹配多面体具有指数扩张复杂性的惊人结果。在这里，对于每一个$n=|V|$，我们描述了一个完美的匹配多面体，这是不同的埃德蒙兹的多面体，并定义了一个较弱的概念扩展制定。我们表明，新的多面体有一个弱的扩展配方（WEF）$Q$的多项式大小。对于每个具有$n$顶点的图$G$，我们可以容易地构造目标函数，使得在$Q$上求解所得到的线性规划决定$G$是否具有完美匹配。构造是统一的意义上说，对于每个$n$，一个单一的多面体被定义为类的所有图与$n$节点。该方法扩展到解决多时间优化问题，如加权匹配问题。在这种情况下，对所构造的WEF进行对数（在最优解的权重中）数量的优化。文中所述的方法涉及到编译器的建设，该编译器将给定的算法在一个规定的伪代码到一个多面体。因此，它可以用来构造一个多面体的任何决策问题，{ \bf P}，可以解决一个给定的算法。与Dobkin-Lipton-Reiss和Valiant的早期结果相比，我们的方法允许直接从标准寄存器模型的算法中构建显式线性程序，而无需中间变换。我们应用我们的结果获得多项式上界的非负秩的某些松弛矩阵的语言的成员资格测试{ \bf P/Poly}。
英文摘要：
A perfect matching in an undirected graph $G=(V,E)$ is a set of vertex disjoint edges from $E$ that include all vertices in $V$. The perfect matching problem is to decide if $G$ has such a matching. Recently Rothvoß proved the striking result that the Edmonds' matching polytope has exponential extension complexity. Here for each $n=|V|$ we describe a perfect matching polytope that is different from Edmonds' polytope and define a weaker notion of extended formulation. We show that the new polytope has a weak extended formulation (WEF) $Q$ of polynomial size. For each graph $G$ with $n$ vertices we can readily construct an objective function so that solving the resulting linear program over $Q$ decides whether or not $G$ has a perfect matching. The construction is uniform in the sense that, for each $n$, a single polytope is defined for the class of all graphs with $n$ nodes. The method extends to solve poly time optimization problems, such as the weighted matching problem. In this case a logarithmic (in the weight of the optimum solution) number of optimizations are made over the constructed WEF. The method described in the paper involves construction of a compiler that converts an algorithm given in a prescribed pseudocode into a polytope. It can therefore be used to construct a polytope for any decision problem in {\bf P} which can be solved by a given algorithm. Compared with earlier results of Dobkin-Lipton-Reiss and Valiant our method allows the construction of explicit linear programs directly from algorithms written for a standard register model, without intermediate transformations. We apply our results to obtain polynomial upper bounds on the non-negative rank of certain slack matrices related to membership testing of languages in {\bf P/Poly}.
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