Steiner Forest: A Simplified Better-Than-2 Approximation

TL;DR

The paper advances Steiner Forest approximation by presenting a simplified framework that achieves a $1.994$-approximation, building on the recent $(2-\varepsilon)$-approximation, yet integrating it with classical Steiner techniques. It combines an $\varepsilon$-extended moat-growing primal-dual method, a time-based contraction approach on actively connected components via submodular gain, and autarkic collections (pairs and triples) to contract and resolve residual instances. A canonical collection derived from the optimal solution and a laminar, crossing-free autarkic construction enable a rigorous win-win argument that either yields large excess in the OPT or a significantly improved objective bound. The result unifies ideas from Steiner Tree relative greedy methods with modern primal-dual refinements, providing a cleaner analytical path toward further potential improvements and a more cohesive view of Steiner network design techniques.

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.

Figures (3)

• Figure 1: An instance where the moat-growing algorithm actively connects all vertices, but no collection $\mathscr{S}$ of vertex sets exists with $\operatorname{gain}(\mathscr{S}) > \operatorname{cost}(\mathscr{S})$. The left figure shows the graph $G$ and the edge costs. The demand pairs are $\{ s,t \}, \{a_1, b_1\}, \dots, \{ a_k,b_k\}$. $\operatorname{OPT}$ consists of the red matching and the green edge, and has cost $k+2$. The primal-dual algorithm (classical or $\varepsilon$-extended) returns a solution of cost $2k+1$, consisting of all red and blue edges. The right figure shows $\operatorname{supp}(y)$. In the dual solution obtained from classical moat-growing, we have $y_U=\frac{1}{2}$ for each orange set $U$; the $\varepsilon$-extended algorithm also has a value of $\varepsilon \cdot (k+1)$ for the dual variable corresponding to the light-blue set $V$.
• Figure 2: Summary of Bounds
• Figure 3: An instance of Steiner Forest with demand pairs $\{a_1,b_1\}, \{a_2,b_2\}, \{a_3,b_3\}$.

Theorems & Definitions (65)

• Theorem 1.1
• Definition 1.2: Separation
• Definition 2.1
• Theorem 2.2
• Definition 3.1: Actively Connected Vertices
• proof
• Definition 3.3: Actively Connected Sets
• Definition 3.4: Gain
• Lemma 3.5: Submodularity
• proof