On the symmetry of the Quadratic Assignment Problem through Elementary Landscape Decomposition

When designing meta-heuristic strategies to optimize the quadratic assignment problem (QAP), it is important to take into account the specific characteristics of the instance to be solved. One of the characteristics that has been pointed out as having the potential to affect the performance of optimization algorithms is the symmetry of the distance and flow matrices that form the QAP. In this paper, we further investigate the impact of the symmetry of the QAP on the performance of meta-heuristic algorithms, focusing on local search based methods. The analysis is carried out using the elementary landscape decomposition (ELD) of the problem under the swap neighborhood. First, we study the number of local optima and the relative contribution of the elementary components on a benchmark composed of different types of instances. Secondly, we propose a specific local search algorithm based on the ELD in order to experimentally validate the effects of the symmetry. The analysis carried out shows that the symmetry of the QAP is a relevant feature that influences both the characteristics of the elementary components and the performance of local search based algorithms.IT1244-19, PID2019-106453GA-I00/AEI/10.13039/501100011033, H202

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

Multi-objectivising Combinatorial Optimisation Problems by means of Elementary Landscape Decompositions

In the last decade, many works in combinatorial optimisation have shown that, due to the advances in multi-objective optimisation, the algorithms from this field could be used for solving single-objective problems as well. In this sense, a number of papers have proposed multi-objectivising single-objective problems in order to use multi-objective algorithms in their optimisation. In this paper, we follow up this idea by presenting a methodology for multi-objectivising combinatorial optimisation prob- lems based on elementary landscape decompositions of their objective function. Under this framework, each of the elementary landscapes obtained from the decomposition is considered as an independent objective function to optimise. In order to illustrate this general methodology, we consider four problems from different domains: the quadratic assignment problem and the linear ordering problem (permutation domain), the 0-1 unconstrained quadratic optimisation problem (binary domain), and the frequency assignment problem (integer domain). We implemented two widely known multi-objective algorithms, NSGA-II and SPEA2, and compared their perfor- mance with that of a single-objective GA. The experiments conducted on a large benchmark of instances of the four problems show that the multi-objective algorithms clearly outperform the single-objective approaches. Furthermore, a discussion on the results suggests that the multi-objective space generated by this decomposition enhances the exploration ability, thus permitting NSGA-II and SPEA2 to obtain better results in the majority of the tested instances.TIN2016-78365R IT-609-1

New Knowledge about the Elementary Landscape Decomposition for Solving the Quadratic Assignment Problem

Previous works have shown that studying the characteristics of the Quadratic Assignment Problem (QAP) is a crucial step in gaining knowledge that can be used to design tailored meta-heuristic algorithms. One way to analyze the characteristics of the QAP is to decompose its objective function into a linear combination of orthogonal sub-functions that can be independently studied. In particular, this work focuses on a decomposition approach that has attracted considerable attention: The Elementary Landscape Decomposition (ELD).The main drawback of the ELD is that it does not allow an understandable characterization of what is being measured by each component of the decomposition. Thus, it turns out difficult to design new efficient meta-heuristic algorithms for the QAP based on the ELD. To address this issue, in this work, we delve deeper into the ELD by means of an additional decomposition of its elementary components. Conducted experiments show that the performed analysis may be used to explain the behaviour of ELD-based methods, providing critical information about their potential applications

Problem Understanding through Landscape Theory

In order to understand the structure of a problem we need to measure some features of the problem. Some examples of measures suggested in the past are autocorrelation and fitness-distance correlation. Landscape theory, developed in the last years in the field of combinatorial optimization, provides mathematical expressions to efficiently compute statistics on optimization problems. In this paper we discuss how can we use optimización combinatoria in the context of problem understanding and present two software tools that can be used to efficiently compute the mentioned measures.Ministerio de Economía y Competitividad (TIN2011-28194

Exact computation of the expectation curves of the bit-flip mutation using landscapes theory

Chicano, F., & Alba E. (2011). Exact computation of the expectation curves of the bit-flip mutation using landscapes theory. Proceedings of 13th Annual Genetic and Evolutionary Computation Conference, Dublin, Ireland, July 12-16, 2011. pp. 2027–2034.Bit-flip mutation is a common operation when a genetic algorithm is applied to solve a problem with binary representation. We use in this paper some results of landscapes theory and Krawtchouk polynomials to exactly compute the expected value of the fitness of a mutated solution. We prove that this expectation is a polynomial in p, the probability of flipping a single bit. We analyze these polynomials and propose some applications of the obtained theoretical results.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. This research has been partially funded by the Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project) and the Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

Elementary landscape decomposition of the 0-1 unconstrained quadratic optimization

Journal of Heuristics, 19(4), pp.711-728Landscapes’ theory provides a formal framework in which combinatorial optimization problems can be theoretically characterized as a sum of an especial kind of landscape called elementary landscape. The elementary landscape decomposition of a combinatorial optimization problem is a useful tool for understanding the problem. Such decomposition provides an additional knowledge on the problem that can be exploited to explain the behavior of some existing algorithms when they are applied to the problem or to create new search methods for the problem. In this paper we analyze the 0-1 Unconstrained Quadratic Optimization from the point of view of landscapes’ theory. We prove that the problem can be written as the sum of two elementary components and we give the exact expressions for these components. We use the landscape decomposition to compute autocorrelation measures of the problem, and show some practical applications of the decomposition.Spanish Ministry of Sci- ence and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

Elementary Components of the Quadratic Assignment Problem

The Quadratic Assignment Problem (QAP) is a well-known NP-hard combinatorial optimization problem that is at the core of many real-world optimization problems. We prove that QAP can be written as the sum of three elementary landscapes when the swap neighborhood is used. We present a closed formula for each of the three elementary components and we compute bounds for the autocorrelation coefficient.Comment: 10 pages, 1 figure. An extended version of this paper was published in GECCO 201

Elementary Landscape Decomposition of the Hamiltonian Path Optimization Problem

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 dis- play this structure are called “Elementary Landscapes” and they have a number of special mathematical properties. The problems that are not elementary landscapes can be decomposed in a sum of elementary ones. This sum is called the elementary landscape decomposition of the problem. In this paper, we provide the elementary landscape decomposi- tion for the Hamiltonian Path Optimization Problem under two different neighborhoods.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Local Optima Networks, Landscape Autocorrelation and Heuristic Search Performance

Chicano, F., Daolio F., Ochoa G., Vérel S., Tomassini M., & Alba E. (2012). Local Optima Networks, Landscape Autocorrelation and Heuristic Search Performance. (Coello, C. A. Coello, Cutello V., Deb K., Forrest S., Nicosia G., & Pavone M., Ed.).Parallel Problem Solving from Nature - PPSN XII - 12th International Conference, Taormina, Italy, September 1-5, 2012, Proceedings, Part II. 337–347.Recent developments in fitness landscape analysis include the study of Local Optima Networks (LON) and applications of the Elementary Landscapes theory. This paper represents a first step at combining these two tools to explore their ability to forecast the performance of search algorithms. We base our analysis on the Quadratic Assignment Problem (QAP) and conduct a large statistical study over 600 generated instances of different types. Our results reveal interesting links between the network measures, the autocorrelation measures and the performance of heuristic search algorithms.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Spanish Ministry of Science and Innovation and FEDER under contract TIN2011-28194. Andalusian Government under contract P07-TIC-03044. Swiss National Science Foundation for financial support under grant number 200021-124578