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. 1

Tabu search (TS) [21] was first applied to graph coloring by Hertz and de Werra in 1987 [23]. Even today, the original Tabucol algorithm as well as its variants are among the best reference algorithms for general graph coloring. In this paper, we describe a Reinforced TS (RTS) coloring algorithm and show improved results on the well-established Dimacs challenge graphs. The improvement is essentially achieved by employing more informative evaluation functions as well as an adaptive technique for tuning the tabu list. We show that RTS can equal the best-known results for many of the Dimacs graphs while remaining quite simple. 1

(PSA) for solving one of the most studied NP-hard optimization

Abstract. The graph coloring problem consists in assigning colors to the vertices of agiven graph Gsuch that no two adjacent vertices receive the same color and the number of used colors is as small as possible. In this paper, we investigate the graph coloring polytope P (G) defined as the convex hull of feasible solutions to the binary programming formulation of the problem. We remark that P (G) coincides with the stable set polytope of a graph constructed from the complement ¯ G of G. We derive facet-defining inequalities for P (G) from independent sets, odd holes, odd anti-holes and odd wheels in ¯ G. Key words: polyhedral combinatorics, graph coloring, polytopes, facets. 1.

ABSTRACT: In this paper, we consider the problem of scheduling n independent jobs on m uniform parallel machines such that total weighted completion time is minimized. We present two meta-heuristics and two hybrid meta-heuristics to solve this problem. Based on a set of instances, a comparative study has been realized in order to evaluate these approaches.

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.

This paper reviews the current state of the literature surrounding methods for the general graph colouring problem and presents a broad comparison of six high-performance algorithms, each belonging to one of the main algorithmic schemes identified. Unlike many previous computational studies in graph colouring, a large range of both artificially generated and real-world graphs are considered, culminating in over 40,000 individual trials that have consumed more than a decade of computation time in total. The picture painted by the comparison is complex, with each method outperforming all others on at least one occasion; however, general patterns are also observed, particularly with regards to the advantages of hybridising local-search techniques with global-based operators. Keywords: Graph Colouring; Metaheuristics; Combinatorial Optimisation. 1

Abstract Graph vertex coloring is one of the most studied NP-hard combinatorial optimization problems. Given the hardness of the problem, various heuristic algorithms have been proposed for practical graph coloring, based on local search, population-based approaches and hybrid methods. The research in graph coloring heuristics is very active and improved results have been obtained recently, notably for coloring large and very large graphs. This chapter surveys and analyzes graph coloring heuristics with a focus on the most recent advances. 1

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This paper is intended to give a review of metaheuristic and their application to combinatorial optimization problems. This paper comprises a snapshot of the rapid evolution of metaheuristic concepts, their convergence towards a unified framework and the richness of potential application in combinatorial optimization problems. Over the years, combinatorial optimization problems are gaining awareness of the researchers both in scientific as well as industrial world. This paper aims to present a brief survey of different metaheuristic algorithms for solving the combinatorial optimization problems. Basically we have divided the metaheuristic into three broad categories namely trajectory methods, population based methods and hybrid methods. Trajectory methods are those that deal with a single solution. These include simulated annealing, tabu search, variable neighborhood search and greedy randomized adaptive search procedure. Population based methods deal with a set of solutions. These include genetic algorithm, ant colony optimization and particle swarm optimization. Hybrid methods deal with the hybridization of single point search methods and population based methods. These are further categorized into five different types. Finally we conclude the paper by giving some issues which are needed to develop a well performed metaheuristic algorithm.