An Improved Analysis of Greedy for Online Steiner Forest

TL;DR

This paper improves the theoretical understanding of the greedy algorithm for the online Steiner Forest problem, showing it is more competitive than previously proven on a broad class of instances, with a bound of O(log(k) log log(k)).

Contribution

The paper provides a tighter analysis demonstrating that the greedy algorithm has an O(log(k) log log(k)) competitive ratio for a wide class of instances, improving upon the previous O(log^2(k)) bound.

Findings

Greedy algorithm is more competitive than previously thought.

The new bound applies to all previously studied instances.

The analysis introduces novel techniques for bounding the competitive ratio.

Abstract

This paper considers the classic Online Steiner Forest problem where one is given a (weighted) graph $G$ and an arbitrary set of $k$ terminal pairs $\{\{s_1,t_1\},\ldots ,\{s_k,t_k\}\}$ that are required to be connected. The goal is to maintain a minimum-weight sub-graph that satisfies all the connectivity requirements as the pairs are revealed one by one. It has been known for a long time that no algorithm (even randomized) can be better than $\Omega(\log(k))$-competitive for this problem. Interestingly, a simple greedy algorithm is already very efficient for this problem. This algorithm can be informally described as follows: Upon arrival of a new pair $\{s_i,t_i\}$, connect $s_i$ and $t_i$ with the shortest path in the current metric, contract the metric along the chosen path and wait for the next pair. Although simple and intuitive, greedy proved itself challenging to analyze and its competitive ratio is a long-standing open problem in the area of online algorithms. The last progress on this question is due to an elegant analysis by Awerbuch, Azar, and Bartal [SODA~1996], who showed that greedy is $O(\log^2(k))$-competitive. Our main result is to show that greedy is in fact $O(\log(k)\log\log(k))$-competitive on a wide class of instances. In particular, this wide class of instances contains all the instances that were exhibited in the literature until now.