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52
Landscapes and Their Correlation Functions
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an addi ..."
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Cited by 105 (16 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.
Fitness Landscapes
 Appl. Math. & Comput
, 2002
"... . Fitness landscapes are a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space that is equipped with some notion of adjacency, nearness, distance or accessibility, into the real numbe ..."
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Cited by 83 (14 self)
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. Fitness landscapes are a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space that is equipped with some notion of adjacency, nearness, distance or accessibility, into the real numbers. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. This chapter gives a brief overview on recent developments in this area, focusing on "geometrical" properties of landscapes. 1 Introduction The concept of a fitness landscape originated in theoretical biology more than seventy years ago [1]. It can be thought of as a kind of "potential function" underlying the dynamics of evolutionary optimization. Implicit in this idea is both a fitness function f that assigns a fitness value to every possible genotype (or organism), and the arrangement of t...
Combinatorial Landscapes
 SIAM REVIEW
, 2002
"... Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, ne ..."
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Cited by 36 (2 self)
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Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems. A fitness landscape is a mapping from a configuration space into the real numbers. The configuration space is equipped with some notion of adjacency, nearness, distance or accessibility. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the "static" properties of landscapes as well as their influence on the dynamics of adaptation. In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.
The algebraic theory of recombination spaces
, 2000
"... A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "g ..."
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Cited by 35 (16 self)
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A new mathematical representation is proposed for the configuration space structure induced by recombination which we called "Pstructure". It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourierdecomposition of fitness landscapes into a superposition of "elementary landscapes". This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for Pstructures. For binary string recombination the elementary landscapes are exactly the pspin functions (Walsh functions), i.e. the same as the elementary landscapes of the string point mutation spaces (i.e. the hypercube). This supports the notion of a strong homomorphisms between string mutation ...
The Theory of Elementary Landscapes
 Applied Mathematical Letters
, 2002
"... When joined to a stipulated neighborhood digraph, an objective function defined on the solution space of a real combinatorial optimization problem forms a landscape. ..."
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Cited by 22 (8 self)
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When joined to a stipulated neighborhood digraph, an objective function defined on the solution space of a real combinatorial optimization problem forms a landscape.
Amplitude spectra of fitness landscapes
 J. COMPLEX SYSTEMS
, 1998
"... Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying con guration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used f ..."
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Cited by 21 (9 self)
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Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying con guration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used for classi cation and comparison purposes. We consider here three very di erent types of landscapes using both mutation and recombination to de ne the topological structure of the con guration spaces. A reliable procedure for estimating the amplitude spectra is presented. The method is based on certain correlation functions that are easily obtained from empirical studies of the landscapes.
zapproximations
 Journal of Algorithms
, 2001
"... Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ..."
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Cited by 16 (3 self)
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Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ratio may not be very meaningful for example, if the ratio of the worst solution to the best solution is at most some constant ff, then an approximation algorithm with factor ff may in fact yield the worst solution! To overcome this hurdle (among others), several authors have independently suggested the use of a different measure which we call zapproximation. An algorithm is an ff zapproximation if it runs in polynomial time, and produces a solution whose distance from the optimal one is at most ff times the distance between the optimal solution and the worst possible solution. The results known so far about zapproximations are either of the inapproximability type or rather straightforward observations. We design polynomial time algorithms for several fundamental discrete optimization problems, in particular we obtain a zapproximation factor of 1 2 for the directed traveling salesman problem (TSP) (with no triangle inequality assumption). For the undirected TSP this improves to
On the classification of NPcomplete problems in terms of their correlation coefficient
 Discrete Applied Mathematics
"... Local search and its variants simulated annealing and tabu search are very popular metaheuristics to approximatively solve NPhard optimization problems. Several experimental studies in the literature have shown that in practice some problems (e.g. the Traveling Salesman Problem, Quadratic Assignmen ..."
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Cited by 15 (2 self)
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Local search and its variants simulated annealing and tabu search are very popular metaheuristics to approximatively solve NPhard optimization problems. Several experimental studies in the literature have shown that in practice some problems (e.g. the Traveling Salesman Problem, Quadratic Assignment Problem) behave very well with these heuristics, whereas others do not (e.g. the Low Autocorrelation Binary String Problem). The autocorrelation function, introduced by Weinberger, measure the ruggedness of a landscape which is formed by a cost function and a neighborhood. We use a derived parameter, named the autocorrelation coefficient, as a tool to better understand these phenomena. In this paper we mainly study cost functions including penalty terms. Our results can be viewed as a first attempt to theoretically justify why it is often better in practice to enlarge the solution space and add penalty terms than to work solely on feasible solutions. Moreover, some new results as well as p...
Fast Fourier Transform for Fitness Landscapes
, 2001
"... We cast some classes of fitness landscapes as problems in spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group a ..."
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Cited by 13 (2 self)
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We cast some classes of fitness landscapes as problems in spectral analysis on various Cayley graphs. In particular, landscapes derived from RNA folding are realized on Hamming graphs and analyzed in terms of Walsh transforms; assignment problems are interpreted as functions on the symmetric group and analyzed in terms of the representation theory of Sn . We show that explicit computations of the Walsh/Fourier transforms are feasible for landscapes with up to 10 8 configurations using Fast Fourier Transform techniques. We find that the cost function of a linear sum assignment problem involves only the defining representation of the symmetric group, while quadratic assignment problems are superpositions of the representations indexed by the partitions (n), (n  1, 1), (n  2, 2), and (n  2, 1, 1). These correspond to the four smallest eigenvalues of the Laplacian of the Cayley graph obtained from using transpositions as the generating set on Sn .
Landscapes and Effective Fitness
, 2003
"... The concept of a fitness landscape arose in theoretical biology, while that of effective fitness has its origin in evolutionary computation. Both have emerged as useful conceptual tools with which to understand the dynamics of evolutionary processes, especially in the presence of complex genotypeph ..."
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Cited by 12 (2 self)
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The concept of a fitness landscape arose in theoretical biology, while that of effective fitness has its origin in evolutionary computation. Both have emerged as useful conceptual tools with which to understand the dynamics of evolutionary processes, especially in the presence of complex genotypephenotype relations. In this contribution we attempt to provide a unified discussion of these two approaches, discussing both their advantages and disadvantages in the context of some simple models. We also discuss how fitness and effective fitness change under various transformations of the configuration space of the underlying genetic model, concentrating on coarse graining transformations and on a particular coordinate transformation that provides an appropriate basis for illuminating the structure and consequences of recombination.