Adaptive-Adversary-Robust Algorithms via Small Copy Tree Embeddings

Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms. Unfortunately, online or dynamic algorithms which use these embeddings seem inherently randomized and ill-suited against adaptive adversaries. In this paper we provide a new tree embedding which addresses these issues by deterministically embedding a graph into a single tree containing O(log n) copies of each vertex while preserving the connectivity structure of every subgraph and O(log² n)-approximating the cost of every subgraph. Using this embedding we obtain the first deterministic bicriteria approximation algorithm for the online covering Steiner problem as well as the first poly-log approximations for demand-robust Steiner forest, group Steiner tree and group Steiner forest.