Problem
Research questions and friction points this paper is trying to address.
Directed Steiner Network
online algorithms
competitive ratio
uncertainty
network design
Hardness of Online Directed Steiner Network
🤖 AI Summary
本文解决了在线定向Steiner网络问题，通过结合代数编码理论，证明了即使对于随机算法和单位成本DAG，也存在信息论上的硬度下界。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
online Directed Steiner Network
information-theoretic hardness
algebraic coding theory
list-recovery bounds
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