A novel particle swarm optimization for the Steiner tree problem in graphs

Abstract

The Steiner tree problem (STP) in graphs is a special but essential case of multiple destination routing (MDR) problems, which focuses on finding a minimal spanning tree (MST) that connecting the source and destinations. It has been proved to be an NP-hard problem. Particle swarm optimization (PSO) is an important swarm intelligent algorithm with fast convergence speed and easy implementation. In this paper, a novel discrete PSO for the STP (DPSO-STP), with the concept that the particle is guided by social and self cognition, is proposed. Different from the standard PSO, the DPSO-STP includes four parts: 1. two preprocessing operations are introduced, which are to construct a complete graph and to calculate each nodepsilas total distance from itself to the source and destination nodes; 2.the position of a particle is represented as a binary string, where 1 stands for the selected nodes and 0 denotes the opposite; 3. several novel update operations, including new mutation factor c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> , are adopted for the binary string; 4. when generating a MST from a binary string, a modified Primpsilas algorithm and a trimming strategy are employed. The experiments based on the benchmarks from category B, C of STP in the OR-library have been carried out to demonstrate the effectiveness of the proposed algorithm. Compared with traditional heuristic algorithms, such as shortest path heuristic (SPH), average distance heuristic (ADH), etc, the DPSO obtains more promising results. And it also performs better than the other iteration based algorithm, with much less computation. The discussion to extend the algorithm to other MDR problems is also given.