A Solution Merging Heuristic for the Steiner Problem in Graphs Using Tree Decompositions

TL;DR

This paper introduces a heuristic for the Steiner Tree Problem that uses tree decompositions to reduce problem complexity, enabling efficient solution merging and improvement on large, sparse graphs.

Contribution

It presents a novel solution merging heuristic leveraging tree decompositions and local search to improve Steiner Tree solutions on large graphs.

Findings

Effective in finding small tree decompositions for large graphs

Often improves solutions quickly using the heuristic

Applicable to sparse benchmark instances

Abstract

Fixed parameter tractable algorithms for bounded treewidth are known to exist for a wide class of graph optimization problems. While most research in this area has been focused on exact algorithms, it is hard to find decompositions of treewidth sufficiently small to make these al- gorithms fast enough for practical use. Consequently, tree decomposition based algorithms have limited applicability to large scale optimization. However, by first reducing the input graph so that a small width tree decomposition can be found, we can harness the power of tree decomposi- tion based techniques in a heuristic algorithm, usable on graphs of much larger treewidth than would be tractable to solve exactly. We propose a solution merging heuristic to the Steiner Tree Problem that applies this idea. Standard local search heuristics provide a natural way to generate subgraphs with lower treewidth than the original instance, and subse- quently we extract an improved solution by solving the instance induced by this subgraph. As such the fixed parameter tractable algorithm be- comes an efficient tool for our solution merging heuristic. For a large class of sparse benchmark instances the algorithm is able to find small width tree decompositions on the union of generated solutions. Subsequently it can often improve on the generated solutions fast.