Dynamic Steiner Tree under Edge Updates
Jeff Giliberti, MohammadTaghi Hajiaghayi, Mohammad Mahdavi, Xinchen Yang
The Steiner tree problem is the canonical generalization of minimum spanning tree (MST). When connectivity is required only among a fixed subset of vertices (terminals), the goal is to maintain a minimum-cost tree connecting these terminals as the graph undergoes edge updates. While dynamic MST has been studied since the 1980s~[Frederickson, STOC'83, SICOMP'85], and the first sublinear-in-$n$ dynamic MST algorithm dates back almost thirty-five years to Eppstein, Galil, Italiano, and Nissenzweig~[FOCS'92, JACM'97], achieving sublinear-in-$n$ update time for dynamic Steiner tree has long remained open, as a single edge update can invalidate many routes in the underlying terminal metric closure. We present the first fully dynamic algorithm for Steiner tree that achieves a constant-factor approximation with sublinear-in-$n$ update time. For any fixed $0<ζ<1/2$, our deterministic algorithm maintains an $O_ζ(1)$-approximate Steiner tree for undirected graphs with polynomially bounded edge weights in $\widetilde O(n^{1/2+ζ})$ amortized update time, answering cost queries in $O(1)$ time, and reporting an explicit tree in $\widetilde O(n^{1+ζ})$ time. Our approach combines periodic terminal-metric rebuilds on a dynamic spanner with a top-tree dynamic program that localizes edge deletions and reconnects surviving components through a dynamic distance sparsifier. Along the way, we give a simpler randomized insertion-only algorithm with a better, explicit approximation ratio.
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