Steiner Forest: A Simplified Better-Than-2 Approximation

TL;DR

This paper presents a simplified and improved approximation algorithm for the Steiner Forest problem, achieving a ratio of 1.994, which is better than the longstanding 2-approximation, by extending recent advanced techniques.

Contribution

It introduces a cleaner, extended approach that combines primal-dual, submodular maximization, and autarkic concepts to improve approximation ratios for Steiner Forest.

Findings

Achieved a 1.994-approximation ratio for Steiner Forest.

Extended and simplified previous advanced algorithms.

Provides a framework for future improvements.

Abstract

In the Steiner Forest problem, we are given a graph with edge lengths, and a collection of demand pairs; the goal is to find a subgraph of least total length such that each demand pair is connected in this subgraph. For over twenty years, the best approximation ratio known for the problem was a $2$-approximation due to Agrawal, Klein, and Ravi (STOC 1991), despite many attempts to surpass this bound. Finally, in a recent breakthrough, Ahmadi, Gholami, Hajiaghayi, Jabbarzade, and Mahdavi (FOCS 2025) gave a $2-\varepsilon$-approximation, where $\varepsilon \approx 10^{-11}$. In this work, we show how to simplify and extend the work of Ahmadi et al. to obtain an improved $1.994$-approximation. We combine some ideas from their work (e.g., an extended run of the moat-growing primal-dual algorithm, and identifying autarkic pairs) with other ideas -- submodular maximization to find components to contract, as in the relative greedy algorithms for Steiner tree, and the use of autarkic triples. We hope that our cleaner abstraction will open the way for further improvements.