$\text{MAX}$–$\text{MIN}$ Ant System

Abstract

Ant System, the first Ant Colony Optimization algorithm, showed to be a viable method for attacking hard combinatorial optimization problems. Yet, its performance, when compared to more fine-tuned algorithms, was rather poor for large instances of traditional benchmark problems like the Traveling Salesman Problem. To show that Ant Colony Optimization algorithms could be good alternatives to existing algorithms for hard combinatorial optimization problems, recent research in this area has mainly focused on the development of algorithmic variants which achieve better performance than Ant System.

In this paper, we present $\text{MAX}$–$\text{MIN}$ Ant System ($\text{MM}\text{AS}$), an Ant Colony Optimization algorithm derived from Ant System. $\text{MM}\text{AS}$ differs from Ant System in several important aspects, whose usefulness we demonstrate by means of an experimental study. Additionally, we relate one of the characteristics specific to $\text{MM}\text{AS}$ — that of using a greedier search than Ant System — to results from the search space analysis of the combinatorial optimization problems attacked in this paper. Our computational results on the Traveling Salesman Problem and the Quadratic Assignment Problem show that $\text{MM}\text{AS}$ is currently among the best performing algorithms for these problems.

Introduction

Ant Colony Optimization (ACO) [8], [11], [13], [14] is a recently developed, population-based approach which has been successfully applied to several $\text{NP}$-hard combinatorial optimization problems [5], [7], [12], [19], [20], [27], [32], [41] (see [10], [11] for an overview). As the name suggests, ACO has been inspired by the behavior of real ant colonies, in particular, by their foraging behavior. One of its main ideas is the indirect communication among the individuals of a colony of agents, called (artificial) ants, based on an analogy with trails of a chemical substance, called pheromone, which real ants use for communication. The (artificial) pheromone trails are a kind of distributed numeric information (called stigmergic information in [9]) which is modified by the ants to reflect their experience accumulated while solving a particular problem. Recently, the ACO metaheuristic has been proposed to provide a unifying framework for most applications of ant algorithms [10], [11] to combinatorial optimization problems. Algorithms which actually are instantiations of the ACO metaheuristic will be called ACO algorithms in the following.

The first ACO algorithm, called Ant System (AS) [8], [13], [14], was applied to the Traveling Salesman Problem (TSP). It gave encouraging results, yet its performance was not competitive with state-of-the-art algorithms for the TSP. Therefore, one important focus of research on ACO algorithms has been the introduction of algorithmic improvements to achieve a much better performance. Typically, these improved algorithms have been tested again on the TSP [6], [12], [42]. While they differ mainly in specific aspects of the search control, all these ACO algorithms are based on a stronger exploitation of the search history to direct the ants’ search process. Recent research on the search space characteristics of some combinatorial optimization problems has shown that for many problems there exists a correlation between the solution quality and the distance from very good or optimal solutions [3], [4], [23], [31]. Hence, it seems reasonable to assume that the concentration of the search around the best solutions found during the search is the key aspect that led to the improved performance shown by the modified ACO algorithms.

The $\text{MAX}$–$\text{MIN}$ Ant System ($\text{MM}\text{AS}$) algorithm discussed in this paper achieves a strong exploitation of the search history by allowing only the best solutions to add pheromone during the pheromone trail update. Also, the use of a rather simple mechanism for limiting the strengths of the pheromone trails effectively avoids premature convergence of the search. Finally, $\text{MM}\text{AS}$ can easily be extended by adding local search algorithms. In fact, the best performing ACO algorithms for many different combinatorial optimization problems improve the solutions generated by the ants with local search algorithms [5], [12], [19], [28], [41], [42], [43]. As our empirical results show, $\text{MM}\text{AS}$ is currently one of the best performing ACO algorithms for the TSP and the Quadratic Assignment Problem (QAP).

The remainder of this paper is structured as follows. In Section 2, we introduce ACO algorithms and discuss their application to the TSP, using AS as a starting point. Next, we review some results from the search space analysis of the TSP which show that solution quality and distance from a global optimum are tightly correlated and we give new results for a similar analysis of the QAP search space. In Section 4, we give details on the modifications of AS leading to $\text{MM}\text{AS}$ and present an experimental investigation showing the effectiveness of these modifications. Section 5 gives the results of our extensive experimental analysis of $\text{MM}\text{AS}$ with additional local search for the TSP. In Section 6, we show that $\text{MM}\text{AS}$ is one of the best available algorithms for the QAP. In Section 7, we briefly summarize our main results and point out directions for further research.