A Comparison of Two Simulated Annealing Algorithms Applied to the Directed Steiner Problem on Networks

Abstract

The well-known Steiner problem on networks is an NP-complete problem for which there are many deterministic algorithms and heuristics. In this paper a new approach to the directed version of this problem is made by applying the ideas of statistical mechanics through the use of the method of simulated annealing. Two different types of cooling algorithms are tailored to the Directed Steiner Problem. Computations are done on a test bed of 480 random graphs in order to study the performance of these algorithms. In addition to the quality of the final answers, the study looks at the distribution of first occurrences during the execution of the cooling schedules of answers that are within 3% of the optimum. The study suggests that these near-optimal answers appear according to an approximately exponential distribution and that they can be obtained in polynomial time. Experiments also indicate that for random graphs annealing usually gives near-optimal answers in less time than does branch and bound. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

TTT plots: a perl program to create time-to-target plots

“…genetic algorithms, iterated local search, tabu search, WalkSAT, and GRASP [4,5,11,12,19,23,31,38,43,47], the random variable time to target solution value is exponentially distributed or fits a two-parameter shifted exponential distribution, i.e., the probability of not having found a given target solution value in t time units is given by P(t) = e −(t−µ)/λ , with λ ∈ R + and µ ∈ R. Hoos and Stützle [22,23] conjecture that this is true for all local search based methods for combinatorial optimization.…”

A robust and scalable algorithm for the Steiner problem in graphs

“…The minimum spanning tree (MST) of the subgraph of G induced by V S (which we denote by MST(G[V S ])) costs no more than S. In particular, if S is optimal, so is MST(G[V S ]). Uchoa and Werneck use dynamic graph techniques to efficiently evaluate neighborhoods defined by the insertion or removal of a single Steiner vertex [24,25,37,43]. Although such neighborhoods had been used in metaheuristics before [1,33,34], they were much slower: O(|V | 2 ) time for insertions and O(|E||V |) time for removals The other two neighborhoods represent a solution S in terms of its key vertices K S , which are Steiner vertices with degree at least three in S. If S is optimal, it costs the same as the MST of its distance network restricted to K S ∪ T (the complete graph on |K S ∪ T | vertices whose edge lengths reflect shortest paths in G).…”

GRASP: basic components and enhancements

“…The speedup is given by the ratio between the times needed to find x, using respectively the sequential algorithm and the parallel implementation with p processors. For a number of metaheuristics, these speedups are linear, such as for example simulated annealing [45,103] and tabu search, provided that the search starts from a local optimum [29,138].…”