Quasi-elementary Landscapes and Superpositions of Elementary Landscapes

Whitley, D., & Chicano F. (2012). Quasi-elementary Landscapes and Superpositions of Elementary Landscapes. (Hamadi, Y., & Schoenauer M., Ed.).Learning and Intelligent Optimization - 6th International Conference, LION 6, Paris, France, January 16-20, 2012, Revised Selected Papers. 277–291.There exist local search landscapes where the evaluation function is an eigenfunction of the graph Laplacian that corresponds to the neighborhood structure of the search space. Problems that display this structure are called “Elementary Landscapes” and they have a number of special mathematical properties. The term “Quasi-elementary landscapes” is introduced to describe landscapes that are “almost” elementary; in quasi-elementary landscapes there exists some efficiently computed “correction” that captures those parts of the neighborhood structure that deviate from the normal structure found in elementary landscapes. The “shift” operator, as well as the “3-opt” operator for the Traveling Salesman Problem landscapes induce quasi-elementary landscapes. A local search neighborhood for the Maximal Clique problem is also quasi-elementary. Finally, we show that landscapes which are a superposition of elementary landscapes can be quasi-elementary in structure.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-11-1-0088. Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

A methodology to find the elementary landscape decomposition of combinatorial optimization problems

Evolutionary Computation Journal 19(4):597-637A small number of combinatorial optimization problems have search spaces that correspond to elementary landscapes, where the objective function f is an eigenfunction of the Laplacian that describes the neighborhood structure of the search space. Many problems are not elementary; however, the objective function of a combinatorial optimization problem can always be expressed as a superposition of multiple elementary landscapes if the underlying neighborhood used is symmetric. This paper presents theoretical results that provide the foundation for algebraic methods that can be used to decompose the objective function of an arbitrary combinatorial optimization problem into a sum of subfunctions, where each subfunction is an elementary landscape. Many steps of this process can be automated, and indeed a software tool could be developed that assists the researcher in finding a landscape decomposition. This methodology is then used to show that the subset sum problem is a superposition of two elementary landscapes, and to show that the quadratic assignment problem is a superposition of three elementary landscapes.Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC- 03044 (DIRICOM project). Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422

Exact computation of the expectation curves for uniform crossover

Chicano, F., Whitley D., & Alba E. (2012). Exact computation of the expectation curves for uniform crossover. (Soule, T., & Moore J. H., Ed.).Genetic and Evolutionary Computation Conference, GECCO'12, Philadelphia, PA, USA, July 7-11, 2012. 1301–1308.Uniform crossover is a popular operator used in genetic algorithms to combine two tentative solutions of a problem represented as binary strings. We use the Walsh decomposition of pseudo-Boolean functions and properties of Krawtchouk matrices to exactly compute the expected value for the fitness of a child generated by uniform crossover from two parent solutions. We prove that this expectation is a polynomial in , the probability of selecting the best-parent bit. We provide efficient algorithms to compute this polynomial for ONEMAX and MAX-kSAT problems, but the results also hold for domains such as NK-Landscapes.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Spanish Ministry of Science and Innovation and FEDER under contract TIN2011-28194 (the roadME project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

10361 Abstracts Collection and Executive Summary -- Theory of Evolutionary Algorithms

From September 5 to 10, the Dagstuhl Seminar 10361 ``Theory of Evolutionary Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general

Dynastic Potential Crossover Operator

An optimal recombination operator for two parent solutions provides the best solution among those that take the value for each variable from one of the parents (gene transmission property). If the solutions are bit strings, the offspring of an optimal recombination operator is optimal in the smallest hyperplane containing the two parent solutions. Exploring this hyperplane is computationally costly, in general, requiring exponential time in the worst case. However, when the variable interaction graph of the objective function is sparse, exploration can be done in polynomial time. In this paper, we present a recombination operator, called Dynastic Potential Crossover (DPX), that runs in polynomial time and behaves like an optimal recombination operator for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with traditional crossover operators, like uniform crossover and network crossover, and with two recently defined efficient recombination operators: partition crossover and articulation points partition crossover. The empirical comparison uses NKQ Landscapes and MAX-SAT instances. DPX outperforms the other crossover operators in terms of quality of the offspring and provides better results included in a trajectory and a population-based metaheuristic, but it requires more time and memory to compute the offspring.This research is partially funded by the Universidad de M\'alaga, Consejería de Economía y Conocimiento de la Junta de Andalucía and FEDER under grant number UMA18-FEDERJA-003 (PRECOG); under grant PID 2020-116727RB-I00 (HUmove) funded by MCIN/AEI/10.13039/501100011033; and TAILOR ICT-48 Network (No 952215) funded by EU Horizon 2020 research and innovation programme. The work is also partially supported in Brazil by São Paulo Research Foundation (FAPESP), under grants 2021/09720-2 and 2019/07665-4, and National Council for Scientific and Technological Development (CNPq), under grant 305755/2018-8

Elementary landscape decomposition of the frequency assignment problem

Theoretical Computer Science 412(43),2011, pp.6002-6019The Frequency Assignment Problem (FAP) is an important problem that arises in the design of radio networks, when a channel has to be assigned to each transceiver of the network. This problem is a generalization of the graph coloring problem. In this paper we study a general version of the FAP that can include adjacent frequency constraints. Using concepts from landscapes’ theory, we prove that this general FAP can be expressed as a sum of two elementary landscapes. Further analysis also shows that some subclasses of the problem correspond to a single elementary landscape. This allows us to compute the kind of neighborhood information that is normally associated with elementary landscapes. We also provide a closed form formula for computing the autocorrelation coefficient for the general FAP, which can be useful as an a priori indicator of the performance of a local search method.This work has been partially funded by the Spanish Ministry of Science and Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project). Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422

Fitness function distributions over generalized search neighborhoods in the q-ary hypercube

Evolutionary Computation, 21(4): 561-590, 2013The frequency distribution of a fitness function over regions of its domain is an important quantity for understanding the behavior of algorithms that employ randomized sampling to search the function. In general, exactly characterizing this distribution is at least as hard as the search problem, since the solutions typically live in the tails of the distribution. However, in some cases it is possible to efficiently retrieve a collection of quantities (called moments) that describe the distribution. In this paper, we consider functions of bounded epistasis that are defined over length-n strings from a finite alphabet of cardinality q. Many problems in combinatorial optimization can be specified as search problems over functions of this type. Employing Fourier analysis of functions over finite groups, we derive an efficient method for computing the exact moments of the frequency distribution of fitness functions over Hamming regions of the q-ary hypercube. We then use this approach to derive equations that describe the expected fitness of the offspring of any point undergoing uniform mutation. The results we present provide insight into the statistical structure of the fitness function for a number of combinatorial problems. For the graph coloring problem, we apply our results to efficiently compute the average number of constraint violations that lie within a certain number of steps of any coloring. We derive an expression for the mutation rate that maximizes the expected fitness of an offspring at each fitness level. We also apply the results to the slightly more complex frequency assignment problem, a relevant application in the domain of the telecommunications industry. As with the graph coloring problem, we provide formulas for the average value of the fitness function in Hamming regions around a solution and the expectation-optimal mutation rate.Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project). Air Force Office of Scientific Re- search, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422

A Comparison of Genetic Algorithms for the Static Job Shop Scheduling Problem

A variety of Genetic Algorithms for the static Job Shop Scheduling Problem have been developed using various methods: direct vs. indirect representations, pure vs. hybrid GA's and serial vs. parallel GA's. We implement a hybrid GA, called OBGT, for solving the static Job Shop Scheduling Problem. A chromosome representation containing the schedule itself is used and order-based crossover operators are combined with techniques that produce active and non-delay schedules