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23
Evolutionary Algorithms and the Maximum Matching Problem
, 2002
"... Randomized search heuristics like evolutionary algorithms are mostly applied to problems whose structure is not completely known but also to combinatorial optimization problems. Practitioners report surprising successes but almost no results with theoretically wellfounded analyses exist. Such a ..."
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Cited by 63 (10 self)
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Randomized search heuristics like evolutionary algorithms are mostly applied to problems whose structure is not completely known but also to combinatorial optimization problems. Practitioners report surprising successes but almost no results with theoretically wellfounded analyses exist. Such an analysis is started in this paper for a fundamental evolutionary algorithm and the wellknown maximum matching problem. It is
Upper and Lower Bounds for Randomized Search Heuristics . . .
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2004
"... Randomized search heuristics like local search, tabu search, simulated annealing or all kinds of evolutionary algorithms have many applications. However, for most problems the best worstcase expected run times are achieved by more problemspecific algorithms. This raises ..."
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Cited by 41 (5 self)
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Randomized search heuristics like local search, tabu search, simulated annealing or all kinds of evolutionary algorithms have many applications. However, for most problems the best worstcase expected run times are achieved by more problemspecific algorithms. This raises
An Analysis of the (µ+1) EA on Simple PseudoBoolean Functions (Extended Abstract)
"... Carsten Witt FB Informatik, LS 2 Univ. Dortmund 44221 Dortmund, Germany carsten.witt@cs.unidortmund.de Abstract. Evolutionary Algorithms (EAs) are successfully applied for optimization in discrete search spaces, but theory is still weak in particular for populationbased EAs. Here, a first r ..."
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Cited by 35 (8 self)
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Carsten Witt FB Informatik, LS 2 Univ. Dortmund 44221 Dortmund, Germany carsten.witt@cs.unidortmund.de Abstract. Evolutionary Algorithms (EAs) are successfully applied for optimization in discrete search spaces, but theory is still weak in particular for populationbased EAs. Here, a first rigorous analysis of the (+1) EA on pseudoBoolean functions is presented. For three example functions wellknown from the analysis of the (1+1) EA, bounds on the expected runtime and success probability are derived. For two of these functions, upper and lower bounds on the expected runtime are tight, and the (+1) EA is never more e#cient than the (1+1) EA. Moreover, all lower bounds grow with . On a more complicated function, however, a small increase of provably decreases the expected runtime drastically.
How To Analyse Evolutionary Algorithms
, 2002
"... Many variants of evolutionary algorithms have been designed and applied. The ..."
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Cited by 26 (1 self)
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Many variants of evolutionary algorithms have been designed and applied. The
Expected Runtimes of a Simple Multiobjective Evolutionary Algorithm
, 2003
"... The expected runtime of a simple multiobjective evolutionary algorithm for the Boolean decision space is analyzed. The algorithm uses independent bit flips as mutation operator and, therefore, searches globally. It is proved that the expected runtime is O(n n) for all objective functions {0, 1} n ..."
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Cited by 21 (3 self)
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The expected runtime of a simple multiobjective evolutionary algorithm for the Boolean decision space is analyzed. The algorithm uses independent bit flips as mutation operator and, therefore, searches globally. It is proved that the expected runtime is O(n n) for all objective functions {0, 1} n → m. This worstcase bound is tight and matches the worstcase bounds for fundamental evolutionary algorithms working in the scenario of singleobjective optimization. For the bicriteria problem LOTZ (Leading Ones Trailing Zeroes), it is shown that the expected runtime is O(n³). Moreover, the runtime is O(n³) with an overwhelming probability. Finally, the function x ↦ → (x 2, (x − 2) 2) that serves as a test function in the continuous decision space is adapted to the Boolean decision space, and bounds on the runtime are derived.
On the Optimization of Monotone Polynomials by Simple Randomized Search Heuristics
, 2002
"... 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 analyze their behavior on classe ..."
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Cited by 17 (9 self)
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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 analyze their behavior on classes of functions. Such an analysis is performed here for the class of monotone pseudoboolean 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.
Complexity Theory and the No Free Lunch Theorem
, 2005
"... Introduction This tutorial reviews basic concepts in complexity theory, as well as various No Free Lunch results and how these results relate to computational complexity. The tutorial explain basic concepts in an informal fashion that illuminates key concepts. "No Free Lunch" theorems for ..."
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Cited by 13 (0 self)
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Introduction This tutorial reviews basic concepts in complexity theory, as well as various No Free Lunch results and how these results relate to computational complexity. The tutorial explain basic concepts in an informal fashion that illuminates key concepts. "No Free Lunch" theorems for search can be summarized by the following result: another when its performance is averaged over all possible discrete functions. Note that "No Free Lunch" is often referred to simply as NFL within the heuristic search community (despite copyrights and trademarks held by the National Football League). No Free Lunch relates to complexity theory in as much as complexity theory addresses the time and space costs of algorithms; complexity theory is also concerned with key classes of problems, such as the class of NP Complete problems that are also of interest to researchers designing search algorithms. 2. Complexity, P and NP The complexity classes denoted by P and NP are the most famous (or notor
Population size vs. runtime of a simple EA
 In Proceedings of the 2003 IEEE Congress on Evolutionary Computation (CEC ’03
, 1996
"... Abstract Evolutionary algorithms (EAs) find numerous applications, and practical knowledge on EAs is immense. In practice, sophisticated populationbased EAs employing selection, mutation and crossover are applied. In contrast, theoretical analysis of EAs often concentrates on very simple algorithm ..."
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Cited by 6 (1 self)
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Abstract Evolutionary algorithms (EAs) find numerous applications, and practical knowledge on EAs is immense. In practice, sophisticated populationbased EAs employing selection, mutation and crossover are applied. In contrast, theoretical analysis of EAs often concentrates on very simple algorithms like the (1+1) EA, where the population size equals 1. In this paper, the question is addressed whether the use of a population by itself can he advantageous. A populationbased EA that does neither make use of crossover nor any diversitymaintaining operator is investigated on an example function. It is shown that an increase of the population size by a polynomial factor decreases the expected runtime from exponential to polynomial. Thereby, the so far best known gap is improved from snperpolynomial to exponential. Moreover, it is proved that the stated runtime bounds occur with a probability exponentially close to one. Finally, a second example function is presented, where opposite results hold. 1
On the optimization of monotone polynomials by the (1+1) EA and randomized local search
 PROC. OF GECCO, LNCS 2723 (2003) 622–633
, 2003
"... 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 analyze their behavior on classes o ..."
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Cited by 6 (1 self)
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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 analyze their behavior on classes of functions. Such an analysis is performed here for the class of monotone pseudoboolean 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.
Running Time Analysis of a Multiobjective Evolutionary Algorithm on Simple and Hard Problems
, 2005
"... In this paper, we suggest a multiobjective evolutionary algorithm based on a restricted mating pool (REMO) with a separate archive for storing the remaining population. Such archive based algorithms have been used for solving realworld applications, however, no theoretical results are available. ..."
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Cited by 5 (0 self)
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In this paper, we suggest a multiobjective evolutionary algorithm based on a restricted mating pool (REMO) with a separate archive for storing the remaining population. Such archive based algorithms have been used for solving realworld applications, however, no theoretical results are available. In this paper, we present a rigorous running time complexity analysis for the algorithm on two simple discrete pseudo boolean functions and on the multiobjective knapsack problem which is known to be NPcomplete. We use two well known simple functions LOTZ (Leading Zeros: Trailing Ones) and a quadratic function. For the knapsack problem we formalize a (1 + ɛ)approximation set under a constraint on the weights of the items. We then generalize the idea by eliminating the constraints based on a principle of partitioning the items into blocks and analyze REMO on it. We use a simple strategy based on partitioning of the decision space into fitness layers for the analysis.