# Online Degree-Bounded Steiner Network Design

## TL;DR

This paper introduces an online algorithm for the degree-bounded Steiner forest problem, achieving an optimal competitive ratio of O(log n), and establishes tight bounds and hardness results for related network design problems.

## Contribution

The paper presents the first online algorithm with an optimal competitive ratio for degree-bounded Steiner forest, advancing understanding of online network design.

## Key findings

- Achieves an O(log n) competitive ratio for the online degree-bounded Steiner forest.
- Proves no algorithm can do better than o(log n) competitive ratio.
- Provides hardness results for related variants like group Steiner tree.

## 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 online 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. We design a simple greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); 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. Fourer and Raghavachari resolved the online variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the natural 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 in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems.