Carsten Witt

Bioinspired computation in combinatorial optimization: algorithms and their computational complexity

Bioinspired computation methods, such as evolutionary algorithms and ant colony optimization, are being applied successfully to complex engineering and combinatorial optimization problems, and it is very important that we understand the computational complexity of these algorithms. This tutorials explains the most important results achieved in this area. The presenters show how runtime behavior can be analyzed in a rigorous way, in particular for combinatorial optimization. They present well-known problems such as minimum spanning trees, shortest paths, maximum matching, and covering and scheduling problems. Classical single objective optimization is examined first. They then investigate the computational complexity of bioinspired computation applied to multiobjective variants of the considered combinatorial optimization problems, and in particular they show how multiobjective optimization can help to speed up bioinspired computation for single-objective optimization problems. The tutorial is based on a book written by the authors with the same title. Further information about the book can be found at www.bioinspiredcomputation.com.

Worst-case and average-case approximations by simple randomized search heuristics

In recent years, probabilistic analyses of algorithms have received increasing attention. Despite results on the average-case complexity and smoothed complexity of exact deterministic algorithms, little is known about the average-case behavior of randomized search heuristics (RSHs). In this paper, two simple RSHs are studied on a simple scheduling problem. While it turns out that in the worst case, both RSHs need exponential time to create solutions being significantly better than 4/3-approximate, an average-case analysis for two input distributions reveals that one RSH is convergent to optimality in polynomial time. Moreover, it is shown that for both RSHs, parallel runs yield a PRAS.

Runtime Analysis of the ( μ +1) EA on Simple Pseudo-Boolean Functions

Although Evolutionary Algorithms (EAs) have been successfully applied to optimization in discrete search spaces, theoretical developments remain weak, in particular for population-based EAs. This paper presents a first rigorous analysis of the (μ+1) EA on pseudo-Boolean functions. Using three well-known example functions from the analysis of the (1+1) EA, we derive bounds on the expected runtime and success probability. For two of these functions, upper and lower bounds on the expected runtime are tight, and on all three functions, the (μ+1) EA is never more efficient than the (1+1) EA. Moreover, all lower bounds grow with μ. On a more complicated function, however, a small increase of μ provably decreases the expected runtime drastically.This paper develops a new proof technique that bounds the runtime of the (μ+1) EA. It investigates the stochastic process for creating family trees of individuals; the depth of these trees is bounded. Thereby, the progress of the population towards the optimum is captured. This new technique is general enough to be applied to other population-based EAs.

Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation

Drift analysis is a powerful tool used to bound the optimization time of evolutionary algorithms (EAs). Various previous works apply a drift theorem going back to Hajek in order to show exponential lower bounds on the optimization time of EAs. However, this drift theorem is tedious to read and to apply since it requires two bounds on the moment-generating (exponential) function of the drift. A recent work identifies a specialization of this drift theorem that is much easier to apply. Nevertheless, it is not as simple and not as general as possible. The present paper picks up Hajek’s line of thought to prove a drift theorem that is very easy to use in evolutionary computation. Only two conditions have to be verified, one of which holds for virtually all EAs with standard mutation. The other condition is a bound on what is really relevant, the drift. Applications show how previous analyses involving the complicated theorem can be redone in a much simpler and clearer way. In some cases even improved results may be achieved. Therefore, the simplified theorem is also a didactical contribution to the runtime analysis of EAs.

Bioinspired Computation in Combinatorial Optimization

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Tight Bounds on the Optimization Time of a Randomized Search Heuristic on Linear Functions

The analysis of randomized search heuristics on classes of functions is fundamental to the understanding of the underlying stochastic process and the development of suitable proof techniques. Recently, remarkable progress has been made in bounding the expected optimization time of a simple evolutionary algorithm, called (1+1) EA, on the class of linear functions. We improve the previously best known bound in this setting from (1.39+ o (1)) en ln n to en ln n + O ( n ) in expectation and with high probability, which is tight up to lower-order terms. Moreover, upper and lower bounds for arbitrary mutation probabilities p are derived, which imply expected polynomial optimization time as long as p = O ((ln n )/ n ) and p = Ω( n − C ) for a constant C > 0, and which are tight if p = c / n for a constant c > 0. As a consequence, the standard mutation probability p = 1/ n is optimal for all linear functions, and the (1+1) EA is found to be an optimal mutation-based algorithm. Furthermore, the algorithm turns out to be surprisingly robust since the large neighbourhood explored by the mutation operator does not disrupt the search.

Approximating covering problems by randomized search heuristics using multi-objective models

The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical ones on this subject. We consider the approximation ability of randomized search heuristics for the class of covering problems and compare single-objective and multi-objective models for such problems. For the Vertex-Cover problem, we point out situations where the multi-objective model leads to a fast construction of optimal solutions while in the single-objective case even no good approximation can be achieved within expected polynomial time. Examining the more general Set-Cover problem we show that optimal solutions can be approximated within a factor of log n, where n is the problem dimension, using the multi-objective approach while the approximation quality obtainable by the single-objective approach in expected polynomial time may be arbitrarily bad.

Ant Colony Optimization and the minimum spanning tree problem

Ant Colony Optimization (ACO) is a kind of metaheuristic 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 metaheuristic 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 algorithm for a combinatorial optimization problem. In our investigations, we consider the minimum spanning tree (MST) 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 Broders algorithm 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 an MST, if the influence of the heuristic information is large enough.

Analysis of diversity-preserving mechanisms for global exploration*

Maintaining diversity is important for the performance of evolutionary algorithms. Diversity-preserving mechanisms can enhance global exploration of the search space and enable crossover to find dissimilar individuals for recombination. We focus on the global exploration capabilities of mutation-based algorithms. Using a simple bimodal test function and rigorous runtime analyses, we compare well-known diversity-preserving mechanisms like deterministic crowding, fitness sharing, and others with a plain algorithm without diversification. We show that diversification is necessary for global exploration, but not all mechanisms succeed in finding both optima efficiently. Our theoretical results are accompanied by additional experiments for different population sizes.

On the Optimization of Monotone Polynomials by Simple Randomized Search Heuristics

Randomized search heuristics like evolutionary algorithms and simulated annealing find many applications, especially in situations where no full information on the problem instance is available. In order to understand how these heuristics work, it is necessary to analyse their behaviour on classes of functions. Such an analysis is performed here for the class of monotone pseudo-Boolean polynomials. Results depending on the degree and the number of terms of the polynomial are obtained. The class of monotone polynomials is of special interest since simple functions of this kind can have an image set of exponential size, improvements can increase the Hamming distance to the optimum and, in order to find a better search point, it can be necessary to search within a large plateau of search points with the same fitness value.