Hardness of Online Directed Steiner Network
TL;DR
This work gives an $\exp\!\bigl(\Omega(\sqrt{\log n})\bigr)$ bound on the competitive ratio, which holds even for randomized algorithms against an oblivious adversary, and on unit-cost DAGs.
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 $\exp\!\bigl(\Omega(\sqrt{\log n})\bigr)$ 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.