关于生成树多面体扩张复杂度的组合下界
On the Combinatorial Lower Bound for the Extension Complexity of the Spanning Tree Polytope
摘要：
在组合优化问题多面体扩张的研究中，一个著名的公开问题是关于具有$n$结点的完全图上最小生成树问题的最小扩张公式的大小.最著名的下界是平凡（维数）界，$Ω(n^2)$，最著名的上界是Wong（1980）对尺寸$O(n^3)$的扩展公式（也是Martin，1991）。本文给出了一个支持生成树松弛矩阵的代价为$\log_2(n^2\log n)+O(1)$的不确定通信协议。这意味着组合下界只能将平凡下界提高（至多）$O(\log n)$倍。
英文摘要：
In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with $n$ nodes. The best known lower bound is the trival (dimension) bound, $Ω(n^2)$, the best known upper bound is the extended formulation by Wong (1980) of size $O(n^3)$ (also Martin, 1991). In this note we give a nondeterministic communication protocol with cost $\log_2(n^2\log n)+O(1)$ for the support of the spanning tree slack matrix. This means that the combinatorial lower bounds can improve the trivial lower bound only by a factor of (at most) $O(\log n)$.
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。
