Online degree-bounded Steiner network design

Abstract

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem - which asks for a subgraph with minimum degree that connects a given set of vertices - is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance. The standard techniques for solving degreebounded problems often fall in the category of iterative and dependent rounding techniques. Unfortunately, these rounding methods are inherently difficult to adapt to an online settings since the underlying fractional solution may change dramatically in between the rounding steps. Indeed, this might be the very reason that despite many advances in the online network design paradigm in the past two decades, the natural family of degreebounded problems has remained widely open. In this paper, we design an intuitive greedy-like algorithm that achieves a competitive ratio of O(logn) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(logn); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems. Fiirer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach and its dual analysis, paves the way for solving the online variants of the classical problems in this family of problems.