Solving the Steiner Tree Problem with few Terminals

TL;DR

This paper introduces DS*, an enhanced algorithm for the Steiner tree problem that allows arbitrary heuristics, including linear programming bounds, improving flexibility and performance in solving large instances.

Contribution

The paper extends the Dijkstra-Steiner algorithm to DS*, enabling the use of admissible heuristics and linear programming bounds, with a new correctness proof and practical implementation.

Findings

DS* allows arbitrary heuristics, including LP-based bounds.

The implementation DS* Solve performs competitively on benchmarks.

Admissibility is a weaker condition than consistency for heuristics.

Abstract

The Steiner tree problem is a well-known problem in network design, routing, and VLSI design. Given a graph, edge costs, and a set of dedicated vertices (terminals), the Steiner tree problem asks to output a sub-graph that connects all terminals at minimum cost. A state-of-the-art algorithm to solve the Steiner tree problem by means of dynamic programming is the Dijkstra-Steiner algorithm. The algorithm builds a Steiner tree of the entire instance by systematically searching for smaller instances, based on subsets of the terminals, and combining Steiner trees for these smaller instances. The search heavily relies on a guiding heuristic function in order to prune the search space. However, to ensure correctness, this algorithm allows only for limited heuristic functions, namely, those that satisfy a so-called consistency condition. In this paper, we enhance the Dijkstra-Steiner algorithm and establish a revisited algorithm, called DS*. The DS* algorithm allows for arbitrary lower bounds as heuristics relaxing the previous condition on the heuristic function. Notably, we can now use linear programming based lower bounds. Further, we capture new requirements for a heuristic function in a condition, which we call admissibility. We show that admissibility is indeed weaker than consistency and establish correctness of the DS* algorithm when using an admissible heuristic function. We implement DS* and combine it with modern preprocessing, resulting in an open-source solver (DS* Solve). Finally, we compare its performance on standard benchmarks and observe a competitive behavior.