路径宽度、树和随机嵌入
Pathwidth, trees, and random embeddings
摘要：
我们证明了，对于每一个$k=1,2,...,$，路径宽度$k$的图上的每一个最短路径度量嵌入到随机树上的分布中，对于某些$c=c(k)$，失真最多为$c$。Gupta、Newman、Rabinovich和Sinclair的一个著名猜想指出，对于图的每个次闭族$F$，存在一个常数$c(F)$，使得来自$F$的图上的每个流实例的多商品最大流/最小割间隙至多为$c(F)$。当家族$F$不包含所有的树时，前面的嵌入定理被用来证明这个猜想。
英文摘要：
We prove that, for every $k=1,2,...,$ every shortest-path metric on a graph of pathwidth $k$ embeds into a distribution over random trees with distortion at most $c$ for some $c=c(k)$. A well-known conjecture of Gupta, Newman, Rabinovich, and Sinclair states that for every minor-closed family of graphs $F$, there is a constant $c(F)$ such that the multi-commodity max-flow/min-cut gap for every flow instance on a graph from $F$ is at most $c(F)$. The preceding embedding theorem is used to prove this conjecture whenever the family $F$ does not contain all trees.
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。
