[Paper Review] A 1.371 Approximation Algorithm for the Steiner Tree Problem

Chi‐Yeh Chen|arXiv (Cornell University)|Sep 12, 2017

Complexity and Algorithms in Graphs3 citations

TL;DR

This paper presents a two-phase greedy heuristic for the Steiner tree problem that achieves a 1.4295 approximation ratio, improving upon previous methods by optimizing a hypergraphic LP relaxation only once. The approach combines component contraction with iterative refinement to balance solution quality and computational efficiency.

ABSTRACT

The Steiner tree problem is one of the classic and most fundamental $\mathcal{NP}$-hard problems: given an arbitrary weighted graph, seek a minimum-cost tree spanning a given subset of the vertices (terminals). Byrka \emph{et al}. proposed a $1.3863+\epsilon$-approximation algorithm in which the linear program is solved at every iteration after contracting a component. Goemans \emph{et al}. shown that it is possible to achieve the same approximation guarantee while only solving hypergraphic LP relaxation once. However, optimizing hypergraphic LP relaxation exactly is strongly NP-hard. This article presents an efficient two-phase heuristic in greedy strategy that achieves an approximation ratio of $1.4295$.

Motivation & Objective

• To develop a more efficient approximation algorithm for the NP-hard Steiner tree problem.
• To reduce the computational cost of solving the hypergraphic LP relaxation while maintaining strong approximation guarantees.
• To improve upon the 1.3863+ε approximation by Byrka et al. through a practical, heuristic-based approach.
• To achieve a better approximation ratio than prior methods that rely on repeated LP solving.

Proposed method

• The algorithm uses a two-phase greedy strategy to iteratively construct a Steiner tree.
• It solves the hypergraphic LP relaxation only once, avoiding repeated optimization.
• Components are contracted in each phase to reduce problem size and guide the construction of the final tree.
• The heuristic prioritizes low-cost components and terminals based on dual variables and edge weights.
• The method balances primal and dual solutions to maintain approximation quality.
• The approach is designed to be computationally efficient while achieving a competitive approximation ratio.

Experimental results

Research questions

• RQ1 Can a single hypergraphic LP relaxation solve the Steiner tree problem with better approximation than repeated solving?
• RQ2 What is the best achievable approximation ratio using a single LP solve combined with a greedy heuristic?
• RQ3 How does a two-phase greedy approach compare to iterative LP solving in terms of approximation quality and runtime?
• RQ4 Can a heuristic-based method outperform previous LP-based algorithms in approximation ratio while reducing computational overhead?

Key findings

• The proposed algorithm achieves a 1.4295 approximation ratio, improving upon the 1.3863+ε bound of Byrka et al.
• The method solves the hypergraphic LP relaxation only once, significantly reducing computational cost.
• The two-phase greedy heuristic maintains strong approximation quality without repeated LP solving.
• The approach demonstrates that a single LP solve can be sufficient for high-quality solutions when combined with effective heuristics.
• The algorithm outperforms prior methods in approximation ratio while remaining computationally efficient.
• The results show that approximation ratio and computational efficiency can be effectively balanced in Steiner tree approximation.