AI 中文总结
该研究证明了在线有向斯坦纳网络问题具有指数级竞争比下界，通过代数编码理论建立信息论难度，表明不确定性本身排除了多对数竞争力的在线算法。
AI 中文摘要
在有向斯坦纳网络（DSN）问题中，给定一个有向图以及一组需求$(s_i,t_i)$，要求找到一个廉价的子图来连接每个终端对。在其在线版本中，需求逐个到达，并且必须通过不可撤销地购买边来服务。DSN是网络设计中一个基本的困难问题，在离线和在线设置中都被广泛研究。在离线情况下，它具有超多对数近似的难度。然而，离线难度并不能说明在线算法的情况，在线算法在计算上不受限制。一个悬而未决的问题是，不确定性本身（即需要在不知道未来需求的情况下承诺解决方案）是否排除了具有多对数竞争力的在线算法。在这项工作中，我们展示了第一个这样的无条件、信息论上的难度结果。即，我们给出了竞争比的$\exp\\!\bigl(\Omega(\sqrt{\log n})\bigr)$下界，该下界即使对针对不知情对手的随机算法以及单位成本的有向无环图也成立。我们的证明使用了在线网络设计与代数编码理论之间的新颖联系。我们使用一个隐藏的低次多项式对请求进行编码，其过去的评估不会泄露未来的评估。然后，我们利用列表恢复界来表明，算法在不知道这些未来评估的情况下无法做出廉价可复用的决策。
英文摘要
In the Directed Steiner Network (DSN) problem we are given a directed graph and a set of demands $(s_i,t_i)$, and asked to find a cheap subgraph connecting each terminal pair. In its online version, the demands arrive online and must be served by buying edges irrevocably. DSN is a fundamental hard problem in network design, heavily studied in both the offline and the online setting. Offline, it has a superpolylogarithmic hardness of approximation. However, offline hardness says nothing about online algorithms, which are computationally unrestricted. It has been an open question whether uncertainty itself (needing to commit to a solution without knowing future demands) rules out polylogarithmic-competitive online algorithms. In this work, we show the first such unconditional, information-theoretic hardness. Namely, we give an $\exp\!\bigl(Ω(\sqrt{\log n})\bigr)$ bound on the competitive ratio, which holds even for randomized algorithms against an oblivious adversary, and on unit-cost DAGs. Our proof uses a novel connection between online network design and algebraic coding theory. We encode requests using a hidden low-degree polynomial, whose past evaluations reveal nothing about future ones. We then use list-recovery bounds to show that an algorithm cannot make cheaply reusable decisions without knowing those future evaluations.
