We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs, while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991), showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.

中文翻译：

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关于对称线性规划的幂

我们考虑决定图（或其他关系结构）属性的对称线性程序（LPs）族，因为对于每个大小的图，都有一个定义多面体提升的 LP，它将与图对应的整数点分开属性来自那些对应于没有属性的图。我们表明，这与具有阈值门的对称布尔电路系列等效，最多具有多项式放大。特别是，当我们考虑多项式大小的 LP 时，该模型等效于具有计数的定点逻辑 (FPC) 的非统一版本中的可定义性。FPC 的已知上限和下限适用于非统一版本。特别是，这意味着具有完美匹配的图类具有多项式大小的对称 LP，而我们获得了哈密顿图类的对称 LP 的指数下界。我们将此与之前的结果（Yannakakis 1991）进行比较和对比，表明匹配和 TSP 多面体的任何对称 LP 都具有指数大小。作为一个应用程序，我们确定对于随机、均匀分布的图，多项式大小的对称 LP 与一般布尔电路一样强大。我们说明了这对经过充分研究的种植集团问题的影响。