Marco Chiarandini

Pareto Local Optimum Sets in the Biobjective Traveling Salesman Problem: An Experimental Study

In this article, we study Pareto local optimum sets for the biobjective Traveling Salesman Problem applying straightforward extensions of local search algorithms for the single objective case. The performance of the local search algorithms is illustrated by experimental results obtained for well known benchmark instances and comparisons to methods from literature. In fact, a 3-opt local search is able to compete with the best performing metaheuristics in terms of solution quality. Finally, we also present an empirical study of the features of the solutions found by 3-opt on a set of randomly generated instances. The results indicate the existence of several clusters of near-optimal solutions that are separated by only a few edges.

Experimental Methods for the Analysis of Optimization Algorithms

In operations research and computer science it is common practice to evaluate the performance of optimization algorithms on the basis of computational results, and the experimental approach should follow accepted principles that guarantee the reliability and reproducibility of results. However, computational experiments differ from those in other sciences, and the last decade has seen considerable methodological research devoted to understanding the particular features of such experiments and assessing the related statistical methods. This book consists of methodological contributions on different scenarios of experimental analysis. The first part overviews the main issues in the experimental analysis of algorithms, and discusses the experimental cycle of algorithm development; the second part treats the characterization by means of statistical distributions of algorithm performance in terms of solution quality, runtime and other measures; and the third part collects advanced methods from experimental design for configuring and tuning algorithms on a specific class of instances with the goal of using the least amount of experimentation. The contributor list includes leading scientists in algorithm design, statistical design, optimization and heuristics, and most chapters provide theoretical background and are enriched with case studies. This book is written for researchers and practitioners in operations research and computer science who wish to improve the experimental assessment of optimization algorithms and, consequently, their design.

Stochastic Local Search Algorithms for Graph Set T-colouring and Frequency Assignment

The graph set T-colouring problem (GSTCP) generalises the classical graph colouring problem; it asks for the assignment of sets of integers to the vertices of a graph such that constraints on the separation of any two numbers assigned to a single vertex or to adjacent vertices are satisfied and some objective function is optimised. Among the objective functions of interest is the minimisation of the difference between the largest and the smallest integers used (the span). In this article, we present an experimental study of local search algorithms for solving general and large size instances of the GSTCP. We compare the performance of previously known as well as new algorithms covering both simple construction heuristics and elaborated stochastic local search algorithms. We investigate systematically different models and search strategies in the algorithms and determine the best choices for different types of instance. The study is an example of design of effective local search for constraint optimisation problems.

Mixed Models for the Analysis of Optimization Algorithms

We review linear statistical models for the analysis of computational experiments on optimization algorithms. The models offer the mathematical framework to separate the effects of algorithmic components and instance features included in the analysis. We regard test instances as drawn from a population and we focus our interest not on those single instances but on the whole population. Hence, instances are treated as a random factor. Overall these experimental designs lead to mixed effects linear models. We present both the theory to justify these models and a computational example in which we analyze and comment on several possible experimental designs. The example is a component-wise analysis of local search algorithms for the 2-edge-connectivity augmentation problem. We use standard statistical software to perform the analysis and report the R commands. Data sets and the analysis in SAS are available in an online compendium.

Heuristic solutions to the long-term unit commitment problem with cogeneration plants

We consider a long-term version of the unit commitment problem that spans over one year divided into hourly time intervals. It includes constraints on electricity and heating production as well as on biomass consumption. The problem is of interest for scenario analysis in long-term strategic planning. We model the problem as a large mixed integer programming problem. Two solutions to this problem are of interest but computationally intractable: the optimal solution and the solution derived by market simulation. To achieve good and fast approximations to these two solutions, we design heuristic algorithms, including mixed integer programming heuristics, construction heuristics and local search procedures. Two setups are the best: a relax and fix mixed integer programming approach with an objective function reformulation and a combination of a dispatching heuristic with stochastic local search. The work is developed in the context of the Danish electricity market and the computational analysis is carried out on real-life data.

The balanced academic curriculum problem revisited

The Balanced Academic Curriculum Problem (BACP) consists in assigning courses to teaching terms satisfying prerequisites and balancing the credit course load within each term. The BACP is part of the CSPLib with three benchmark instances, but its formulation is simpler than the problem solved in practice by universities. In this article, we introduce a generalized version of the problem that takes different curricula and professor preferences into account, and we provide a set of real-life problem instances arisen at University of Udine. Since the existing formulation based on a min–max objective function does not balance effectively the credit load for the new instances, we also propose alternative objective functions. Whereas all the CSPLib instances are efficiently solved with Integer Linear Programming (ILP) state-of-the-art solvers, our new set of real-life instances turns out to be much more challenging and still intractable for ILP solvers. Therefore, we have designed, implemented, and analyzed heuristics based on local search. We have collected computational results on all the new instances with the proposed approaches and assessed the quality of solutions with respect to the lower bounds found by ILP on a relaxed and decomposed problem. Results show that a selected heuristic finds solutions of quality at 9%–60% distance from the lower bound. We make all data publicly available, in order to stimulate further research on this problem.

An analysis of heuristics for vertex colouring

Several heuristics have been presented in the literature for finding a proper colouring of the vertices of a graph using the least number of colours. These heuristics are commonly compared on a set of graphs that served two DIMACS competitions. This set does not permit the statistical study of relations between algorithm performance and structural features of graphs. We generate a new set of random graphs controlling their structural features and advance the knowledge of heuristics for graph colouring. We maintain and make all algorithms described here publically available in order to facilitate future comparisons.

Very Large-Scale Neighborhood Search: Overview and Case Studies on Coloring Problems

Summary. Two key issues in local search algorithms are the definition of a neighborhood and the way to examine it. In this chapter we consider techniques for examining very large neighborhoods, in particular, ways for exactly searching them. We first illustrate such techniques using three paradigmatic examples. In the largest part of the chapter, we focus on the development and experimental study of very largescale neighborhood search algorithms for two coloring problems. The first example concerns the well-known (vertex) graph coloring problem. Despite initial promising results on the use of very large-scale neighborhoods, our final conclusion was negative: the usage of the proposed very large-scale neighborhoods did not help to improve the performance of effective stochastic local search algorithms. The second example, the graph set T-coloring problem, yielded more positive results. In this case, a very large-scale neighborhood that was specially tailored for this problem and that can be efficiently searched, resulted to be an essential component of a new state-of-the-art algorithm for various instance classes.

Heuristic algorithms for Hadamard matrices with two circulant cores

We design heuristic algorithms to construct Hadamard matrices with two circulant cores. This hard combinatorial problem can be formulated in terms of objective functions of several binary variables, so that heuristic methodologies can be used. Our algorithms are based on local and tabu search and they use information on the geometry of the objective function landscapes. In addition, we use the supplementary difference sets formalism to detect when solutions of a special structure exist. Using these algorithms we have computed at least one Hadamard matrix with two circulant cores of the sixteen orders 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116. In particular, the Hadamard matrix with two circulant cores of order 116 is constructed here for the first time, indeed it was accidentally reported as known in an earlier paper.

Mixed models for the analysis of local search components

We consider a possible scenario of experimental analysis on heuristics for optimization: identifying the contribution of local search components when algorithms are evaluated on the basis of solution quality attained. We discuss the experimental designs with special focus on the role of the test instances in the statistical analysis. Contrary to previous practice of modeling instances as a blocking factor, we treat them as a random factor. Together with algorithms, or their components, which are fixed factors, this leads naturally to a mixed ANOVA model. We motivate our choice and illustrate the application of the mixed model on a study of local search for the 2-edge-connectivity problem.