Ant Colony Optimization and the Minimum Spanning Tree Problem

Ant Colony Optimization (ACO) is a kind of randomized search heuristic that has become very popular for solving problems from combinatorial optimization. Solutions for a given problem are constructed by a random walk on a so-called construction graph. This random walk can be influenced by heuristic information about the problem. In contrast to many successful applications, the theoretical foundation of this kind of randomized search heuristic is rather weak. Theoretical investigations with respect to the runtime behavior of ACO algorithms have been started only recently for the optimization of pseudo-boolean functions. We present the first comprehensive rigorous analysis of a simple ACO algorithms for a combinatorial optimization problem. In our investigations we consider the minimum spanning tree problem and examine the effect of two construction graphs with respect to the runtime behavior. The choice of the construction graph in an ACO algorithm seems to be crucial for the success of such an algorithm. First, we take the input graph itself as the construction graph and analyze the use of a construction procedure that is similar to Broder’s algorithm [1] for choosing a spanning tree uniformly at random. After that, a more incremental construction procedure is analyzed. It turns out that this procedure is superior to the Broder-based algorithm and produces additionally in a constant number of iterations a minimum spanning tree if the influence of the heuristic information is large enough. Financial support by the Deutsche Forschungsgemeinschaft (SFB) in terms of the Collaborative Research Center “Computational Intelligence” (SFB 531) is gratefully acknowledged. 1 Electronic Colloquium on Computational Complexity, Report No. 143 (2006)