The paper revisits the robust $s$-$t$ path problem, one of the most fundamental problems in robust optimization. In the problem, we are given a directed graph with $n$ vertices and $k$ distinct cost functions (scenarios) defined over edges, and aim to choose an $s$-$t$ path such that the total cost of the path is always provable no matter which scenario is realized. With the view of each cost function being associated with an agent, our goal is to find a common $s$-$t$ path minimizing the maximum objective among all agents, and thus create a fair solution for them. The problem is hard to approximate within $o(\log k)$ by any quasi-polynomial time algorithm unless $\mathrm{NP} \subseteq \mathrm{DTIME}(n^{\mathrm{poly}\log n})$, and the best approximation ratio known to date is $\widetilde{O}(\sqrt{n})$ which is based on the natural flow linear program. A longstanding open question is whether we can achieve a polylogarithmic approximation even when a quasi-polynomial running time is allowed. We give the first polylogarithmic approximation for robust $s$-$t$ path since the problem was proposed more than two decades ago. In particular, we introduce a $O(\log n \log k)$-approximate algorithm running in quasi-polynomial time. The algorithm is built on a novel linear program formulation for a decision-tree-type structure which enables us to get rid of the $\Omega(\max\{k,\sqrt{n}\})$ integrality gap of the natural flow LP. Further, we also consider some well-known graph classes, e.g., graphs with bounded treewidth, and show that the polylogarithmic approximation can be achieved polynomially on these graphs. We hope the new proposed techniques in the paper can offer new insights into the robust $s$-$t$ path problem and related problems in robust optimization.

翻译：本文重新审视了稳健s-t路径问题，这是稳健优化中最基本的问题之一。在该问题中，我们给定一个有n个顶点和k个定义在边上的不同成本函数（情景）的有向图，目标是选择一条s-t路径，使得无论哪个情景实现，该路径的总成本始终可证。将每个成本函数视为一个代理，我们的目标是找到一条公共的s-t路径，最小化所有代理中的最大目标值，从而为其创造公平的解决方案。除非NP ⊆ DTIME(n^{poly log n})，否则任何拟多项式时间算法都难以在o(log k)内近似该问题，而目前已知的最佳近似比为基于自然流线性规划的Õ(√n)。一个长期存在的开放问题是：即使允许拟多项式运行时间，我们是否能够实现多对数近似？自该问题提出二十多年来，我们首次为其给出了多对数近似。特别地，我们提出了一种运行在拟多项式时间内的O(log n log k)近似算法。该算法基于一种针对决策树类型结构的新型线性规划公式，使我们能够避免自然流LP的Ω(max{k,√n})整性间隙。此外，我们还考虑了一些著名的图类（例如有界树宽的图），并表明在这些图上可以通过多项式时间实现多对数近似。我们希望本文提出的新技术能够为稳健s-t路径问题以及稳健优化中的相关问题提供新的见解。