We consider the problem of finding edge-disjoint paths between given pairs of vertices in a sufficiently strong $d$-regular expander graph $G$ with $n$ vertices. In particular, we describe a deterministic, polynomial time algorithm which maintains an initially empty collection of edge-disjoint paths $\mathcal P$ in $G$ and fulfills any series of two types of requests: 1. Given two vertices $a$ and $b$ such that each appears as an endpoint in $O(d)$ paths in $\mathcal P$ and, additionally, $|\mathcal P| = O(n d / \log n)$, the algorithm finds a path of length at most $\log n$ connecting $a$ and $b$ which is edge-disjoint from all other paths in $\mathcal P$, and adds it to $\mathcal P$. 2. Remove a given path $P \in \mathcal{P}$ from $\mathcal{P}$. Importantly, each request is processed before seeing the next one. The upper bound on the length of found paths and the constraints are the best possible up to a constant factor. This establishes the first online algorithm for finding edge-disjoint paths in expanders which also allows removals, significantly strengthening a long list of previous results on the topic.

翻译：考虑在足够强的$d$正则扩张图$G$（$n$个顶点）中寻找给定顶点对间边不交路径的问题。具体地，我们描述一个确定性多项式时间算法，该算法维护$G$中初始为空的边不交路径集合$\mathcal P$，并能处理以下两类请求序列：1. 给定两个顶点$a$和$b$，满足每个顶点在$\mathcal P$中作为端点出现$O(d)$次，且$|\mathcal P| = O(n d / \log n)$，算法找到一条长度至多为$\log n$的路径连接$a$和$b$，该路径与$\mathcal P$中所有其他路径边不交，并将其加入$\mathcal P$；2. 从$\mathcal P$中删除指定路径$P \in \mathcal{P}$。关键之处在于，每个请求均在下个请求到达前处理完毕。所找到路径长度的上界及约束条件在常数因子意义下达到最优。这建立了扩张图中首个支持移除操作的在线边不交路径算法，显著强化了该主题上此前一系列研究成果。