Path-distance heuristics for the Steiner problem in undirected networks

Abstract

Abstract. An integrative overview of the algorithmic characteristics of three well-known polynomialtime heuristics for the undirected Steiner minimum tree problem: shortest path heuristic (SPH), distance network heuristic (DNH), and average distance heuristic (ADH) is given. The performance of these single-pass heuristics (and some variants) is compared and contrasted with several heuristics based on repetitive applications of the SPH. It is shown that two of these repetitive SPH variants generate solutions that in general are better than solutions obtained by any single-pass heuristic. The worst-case time complexity of the two new variants is O(pn 3) and O(p3n2), while the worst-case time complexity of the SPH, DNH, and ADH is respectively 0(pn2), O(m + n log n), and O(n 3) where p is the number of vertices to be spanned, n is the total number of vertices, and m is the total number of edges. However, use of few simple tests is shown to provide large reductions of problem instances (both in terms of vertices and in term of edges). As a consequence, a substantial speed-up is obtained so that the repetitive variants are also competitive with respect to running times.

Computing Near-Optimal Solutions to the Steiner Problem in a Graph Using a Genetic Algorithm

“…Routine graphReductions of Fig. 2 performs four kinds of rather simple reductions all of which are described in [30,31]. More elaborate reductions as well as proofs of the correctness of the reductions used here can be found in [9].…”

“…Therefore, the distance graph D(G) is computed initially using Floyd's algorithm [1] [6]. In [30,31] the following reduction is also suggested along with the reductions described above: If max{c(sp(v, u)), c(sp(v, to))} < c(e uw ), e uw ∈ E and v ∈ W , then no MStT can include e uw , which therefore can be deleted. However, in this case the required update of D(G) has a worst case complexity of O(n 3 ) using Dionne and Florian's algorithm [6].…”

“…Furthermore, reductions of distinct types are mutually dependent in the sense that performing all possible reductions of some type may allow new subsequent reductions of another type. It is not clear in which order reductions should be performed to obtain the overall best reduction of a given graph [31]. The arbitrarily chosen scheme for performing reductions in routine graphReductions is shown in Fig.…”

“…This test suite consists of randomly generated graphs with up to 2,500 vertices and 62,500 edges. The obtained performance is compared to that of the GA by Kapsalis et al [18], an iterated version of the Shortest Path Heuristic called SPH-I, which is one of the very best deterministic heuristics [31], and two recent branch-and-cut algorithms by Lucena and Beasley [23] and Chopra, Gorres and Rao [5]. The experimental results shows the following:…”

Exact algorithms for Steiner tree problem

“…Various algorithms have been proposed for solving STP such as exact algorithms [77,30,13,18,21,48,49,94,97,4,6,57], approximation algorithms [17,7,10,13,102,89,53,56] and heuristic algorithms [81,19,9,71,60,13,92]. This thesis focuses mainly on exact algorithms for the Steiner tree problem.…”