A recent and very promising approach for combinatorial optimization is to embed local search into the framework of evolutionary algorithms. In this paper, we present such hybrid algorithms for the graph coloring problem. These algorithms combine a new class of highly specialized crossover operators and a well-known tabu search algorithm. Experiments of such a hybrid algorithm are carried out on large DIMACS Challenge benchmark graphs. Results prove very competitive with and even better than those of state-of-the-art algorithms. Analysis of the behavior of the algorithm sheds light on ways to further improvement. Keywords: Graph coloring, solution recombination, tabu search, combinatorial optimization. 1 Introduction A recent and very promising approach for combinatorial optimization is to embed local search into the framework of population based evolutionary algorithms, leading to hybrid evolutionary algorithms (HEA). Such an algorithm is essentially based on two key elements: an eff...

We describe a general approach to optimization which we term "Squeaky Wheel" Optimization (swo). In swo, a greedy algorithm is used to construct a solution which is then analyzed to find the trouble spots, i.e., those elements, that, if improved, are likely to improve the objective function score. The results of the analysis are used to generate new priorities that determine the order in which the greedy algorithm constructs the next solution. This Construct/Analyze/Prioritize cycle continues until some limit is reached, or an acceptable solution is found. SWO can be viewed as operating on two search spaces: solutions and prioritizations. Successive solutions are only indirectly related, via the re-prioritization that results from analyzing the prior solution. Similarly, successive prioritizations are generated by constructing and analyzing solutions. This "coupled search" has some interesting properties, which we discuss. We report encouraging experimental results on two ...

. This paper presents a new genetic local search algorithm for the graph coloring problem. The algorithm combines an original crossover based on the notion of union of independent sets and a powerful local search operator (tabu search). This new hybrid algorithm allows us to improve on the best known results of some large instances of the famous Dimacs benchmarks. 1 Introduction The graph coloring problem is one of the most studied NP-hard problems and can be defined informally as follows. Given an undirected graph, one wishes to color with a minimal number of colors the nodes of the graph in such a way that two colors assigned to two adjacent nodes must be different. Graph coloring has many practical applications such as timetabling and resource assignment. Given the NP-completeness of the coloring problem, it becomes natural to design heuristic methods. Indeed many heuristic methods have been developed, constructive methods in the 60's and 70's [1, 12], local search meta-heuristics ...

: In this paper, a generic tabu search is presented for three coloring problems: graph coloring, T-colorings and set T-colorings. This algorithm integrates important features such as greedy initialization, solution re-generation, dynamic tabu tenure, incremental evaluation of solutions and constraint handling techniques. Empirical comparisons show that this algorithm approaches the best coloring algorithms and outperforms some hybrid algorithms on a wide range of benchmarks. Experiments on large random instances of T-colorings and set T-colorings show encouraging results. 3.1 INTRODUCTION The graph coloring problem is one of the most studied NP-hard problems and can be defined informally as follows. Given an undirected graph, one wishes to color the nodes of the graph with a minimal number of colors in such a way that two colors assigned to two adjacent nodes must be di#erent, i.e., have a minimal distance greater than zero. Graph coloring has many practical applications such as timet...

Abstract. We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.

Two contrasting search paradigms for solving combinatorial problems are systematic backtracking and local search. The former is often eective on highly structured problems because of its ability to exploit consistency techniques, while the latter tends to scale better on very large problems. Neither approach is ideal for all problems, and a current trend in arti cial intelligence is the hybridisation of search techniques. This paper describes a use of forward checking in local search: pruning coloration neighbourhoods for graph colouring. The approach is evaluated on standard benchmarks and compared with several other algorithms. Good results are obtained; in particular, one variant nds improved colourings on geometric graphs, while another is very eective on equipartite graphs. Its application to other combinatorial problems is discussed.

Wehavedevelopedanewalgorithmandsoftwareforgraph coloringbysystematicallycombiningseveralalgorithmand softwaredevelopmentideasthathadcrucialimpactonthe algorithm'sperformance.Thealgorithmexploresthedivideand -conquerparadigm,globalsearchforconstrainedindependentsetsusingacomputationallyinexpensiveobjective function,assignmentofmost-constrainedverticestoleastconstrainingcolors, reuseandlocalityexplorationofintermediatesolutions, searchtimemanagement,post-processing lottery-schedulingiterativeimprovement,andstatisticalparameterdeterminationandvalidation. Thealgorithmwas testedonasetofreal-lifeexamples.Wefoundthathard-tocolorreal -lifeexamplesarecommonespeciallyindomains whereproblemmodelingresultsindensergraphs.Systematicexperimentationsdemonstratedthatfornumerousin - stancesthealgorithmoutperformedallotherimplementationsreportedinliteratureinsolutionqualityandrun -time. 1Introduction Theever-increasingamountofresourcesencounteredincontemporarydesignswithexponentiallyascendingcomplexi -...

This paper presents a new neighbourhood structure for Graph Colouring problems based upon permutations of the colours of vertices in a subgraph. This neighbourhood is an exponentially large subspace of the solution space, for which a local optimum can be found in polynomial time using an algorithm called Permutation Descent. We thus provide

We present a diversity-oriented hybrid evolutionary approach for the graph coloring problem. This approach is based on both generally applicable strategies and specifically tailored techniques. Particular attention is paid to ensuring population diversity by carefully controlling spacing among individuals. Using a distance measure between potential solutions, the general population management strategy decides whether an offspring should be accepted in the population, which individual needs to be replaced and when mutation is applied. Furthermore, we introduce a special grouping-based multi-parent crossover operator which relies on several relevant features to identify meaningful building blocks for offspring construction. The proposed approach can be generally characterized as “well-informed”, in the sense that the design of each component is based on the most pertinent information which is identified by both experimental observation and careful analysis of the given problem. The resulting algorithm proves to be highly competitive when it is applied on the whole set of the DIMACS benchmark graphs.

We present a search space analysis and its application in improving local search algorithms for the graph coloring problem. Using a classical distance measure between colorings, we introduce the following clustering hypothesis: the high quality solutions are not randomly scattered in the search space, but rather grouped in clusters within spheres of specific diameter. We first provide intuitive evidence for this hypothesis by presenting a projection of a large set of local minima in the 3D space. An experimental confirmation is also presented: we introduce two algorithms that exploit the hypothesis by guiding an underlying Tabu Search (TS) process. The first algorithm (TS-Div) uses a learning process to guide the basic TS process toward as-yet-unvisited spheres. The second algorithm (TS-Int) makes deep investigations within a bounded region by organizing it as a tree-like structure of connected spheres. We experimentally demonstrate that if such a region contains a global optimum, TS-Int does not fail in eventually finding it. This pair of algorithms significantly outperforms the underlying basic TS algorithm; it can even improve some of the best-known solutions ever reported in the literature (e.g. for dsjc1000.9). Key words: graph coloring, local optima distribution, search by learning. 1