Okamura-Seymour图的最小多割与Steiner森林

路径与交点：有向 Okamura-Seymour 实例中拟度量的刻画

Paths and Intersections: Characterization of Quasi-metrics in Directed Okamura-Seymour Instances

本文研究有向 Okamura-Seymour 图中的拟度量实现问题，证明给定边界终端循环次序时 Monge 性质为充要条件，并给出构造实例及寻找循环次序的高效算法。

Okamura–Seymour 定理的一个近似推广

An Approximate Generalization of the Okamura-Seymour Theorem

本文将 Okamura–Seymour 的单面需求设定推广到每条需求边端点共面的平面图，并借助保持同面点对距离的 L1 嵌入，证明割条件可路由 Ω(1) 比例需求。

路径与交点：冈村-西摩实例的最小实现

Paths and Intersections: Repelling Pairs

研究冈村-西摩实例最短路径度量反问题，给定循环有序终端集上的OS度量D，目标是找到其最小实现，通过分析图结构，证明D确定规范中间图模板，可恢复最小实现的底层嵌入图并计算边长。

Improved Compression of the Okamura-Seymour Metric

Let $G=(V,E)$ be an undirected unweighted planar graph. Consider a vector storing the distances from an arbitrary vertex $v$ to all vertices $S = \{ s_1 , s_2 , \ldots , s_k \}$ of a single face in their cyclic order. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values of this vector. In STOC'19, Li and Parter used a VC-dimension argument to show that in planar graphs, the number of distinct patterns, denoted $x$, is only $O(k^3)$. This resulted in a simple compression scheme requiring $\tilde O(\min \{ k^4+|T|, k\cdot |T|\})$ space to encode the di…

Okamura-Seymour 拟度量的平面嵌入的多项式时间算法及其在分布式 SSSP 中的应用

Planar Embedding of Okamura-Seymour Quasimetrics in Polynomial Time with an Application to Distributed SSSP

提出首个多项式时间算法，将 Okamura-Seymour 拟度量嵌入平面图，并应用于分布式 CONGEST 模型中的平面有向图单源最短路径近似，达到接近最优的轮数复杂度。

Multicommodity Flows in Planar Graphs with Demands on Faces

We consider the problem of multicommodity flows in planar graphs. Seymour showed that if the union of supply and demand graphs is planar, then the cut condition is sufficient for routing demands. Okamura-Seymour showed that if all demands are incident on one face, then again cut condition is sufficient for routing demands. We consider a common generalization of these settings where the end points of each demand are on the same face of the planar graph. We show that if the source sink pairs on each face of the graph are such that sources and sinks appear contiguously on the cycle bounding the f…