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33
Exploring the kcolorable Landscape with Iterated Greedy
 Dimacs Series in Discrete Mathematics and Theoretical Computer Science
, 1995
"... . Many heuristic algorithms have been proposed for graph coloring. The simplest is perhaps the greedy algorithm. Many variations have been proposed for this algorithm at various levels of sophistication, but it is generally assumed that the coloring will occur in a single attempt. We note that if a ..."
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Cited by 42 (3 self)
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. Many heuristic algorithms have been proposed for graph coloring. The simplest is perhaps the greedy algorithm. Many variations have been proposed for this algorithm at various levels of sophistication, but it is generally assumed that the coloring will occur in a single attempt. We note that if a new permutation of the vertices is chosen which respects the independent sets of a previous coloring, then applying the greedy algorithm will result in a new coloring in which the number of colors used does not increase, yet may decrease. We introduce several heuristics for generating new permutations that are fast when implemented and effective in reducing the coloring number. The resulting Iterated Greedy algorithm(IG) can obtain colorings in the range 100 to 103 on graphs in G 1000; 1 2 . More interestingly, it can optimally color kcolorable graphs with k up to 60 and n = 1000. We couple this algorithm with several other coloring algorithms, including a modified TABU search, and one t...
Multilevel Refinement for Combinatorial Optimisation Problems
 SE10 9LS
, 2001
"... 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 (some ..."
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Cited by 28 (5 self)
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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.
On the Futility of Blind Search
 EVOLUTIONARY COMPUTATION
, 1996
"... This paper might have been subtitled "An algorithmicist looks at no free lunch." We use simple adversary arguments to redevelop and explore some ofthenofreelunch (NFL) theorems and perhaps extend them a little. A second goal is to clarify the relationship of NFL theorems to algorithm theor ..."
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Cited by 25 (1 self)
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This paper might have been subtitled "An algorithmicist looks at no free lunch." We use simple adversary arguments to redevelop and explore some ofthenofreelunch (NFL) theorems and perhaps extend them a little. A second goal is to clarify the relationship of NFL theorems to algorithm theory. In particular we claim that NFL puts much weaker restrictions on the claims that an evolutionary algorithm can make than does acceptance of the conjectures of traditional complexity theory. And third we take a brief look at whether the notion of natural evolution relates to optimization, and what if any the implications of evolution are for computing. In this part, we mostly try to raise questions concerning the validity of applying the genetic model to the problem of optimization. This is an informal paper  most of the information presented is not formally proven, and is either "common knowledge" or formally proven elsewhere. Some of the claims are intuitions based on experience with algorithms, and in a more formal setting should be classi ed as conjectures. Thegoalisnotsomuch todevelop theory, asitisto perhaps persuade the reader to adopt a particular viewpoint.
Scalable Parallel Graph Coloring Algorithms
, 2000
"... Finding a good graph coloring quickly is often a crucial phase in the development of efficient, parallel algorithms for many scientific and engineering applications. In this paper we consider the problem of solving the graph coloring problem itself in parallel. We present a simple and fast paral ..."
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Cited by 23 (7 self)
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Finding a good graph coloring quickly is often a crucial phase in the development of efficient, parallel algorithms for many scientific and engineering applications. In this paper we consider the problem of solving the graph coloring problem itself in parallel. We present a simple and fast parallel graph coloring heuristic that is well suited for shared memory programming and yields an almost linear speedup on the PRAM model. We also present a second heuristic that improves on the number of colors used. The heuristics have been implemented using OpenMP. Experiments conducted on an SGI Cray Origin 2000 super computer using very large graphs from finite element methods and eigenvalue computations validate the theoretical runtime analysis.
An effective hybrid algorithm for university course timetabling
, 2006
"... The university course timetabling problem is an optimisation problem in which a set of events has to be scheduled in timeslots and located in suitable rooms. Recently, a set of benchmark instances was introduced and used for an ‘International Timetabling Competition’ to which 24 algorithms were subm ..."
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Cited by 20 (6 self)
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The university course timetabling problem is an optimisation problem in which a set of events has to be scheduled in timeslots and located in suitable rooms. Recently, a set of benchmark instances was introduced and used for an ‘International Timetabling Competition’ to which 24 algorithms were submitted by various research groups active in the field of timetabling. We describe and analyse a hybrid metaheuristic algorithm which was developed under the very same rules and deadlines imposed by the competition and outperformed the official winner. It combines various construction heuristics, tabu search, variable neighbourhood descent and simulated annealing. Due to the complexity of developing hybrid metaheuristics, we strongly relied on an experimental methodology for configuring the algorithms as well as for choosing proper parameter settings. In particular, we used racing procedures that allow an automatic or semiautomatic configuration of algorithms with a good save in time. Our successful example shows that the systematic design of hybrid algorithms through an experimental methodology leads to high performing algorithms for hard combinatorial optimisation problems.
Graph Coloring for Air Traffic Flow Management
 PROCEEDINGS CPAIOR’02
, 2002
"... The aim of Air Traffic Flow Management (ATFM) is to enhance the capacity of the airspace while satisfying Air Traffic Control constraints and airlines requests to optimize their operating costs. This paper presents a design of a new route network that tries to optimize these criteria. The basic idea ..."
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Cited by 18 (2 self)
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The aim of Air Traffic Flow Management (ATFM) is to enhance the capacity of the airspace while satisfying Air Traffic Control constraints and airlines requests to optimize their operating costs. This paper presents a design of a new route network that tries to optimize these criteria. The basic idea is to consider direct routes only and to vertically separate intersecting flows of aircrafts by allocating distinct flight levels. This problem is a graph coloring problem that we tackle using Constraint Programming and a greedy algorithm to find cliques of the constraint graph which are used to post global constraints. Through the search for optimal solutions minimizing the number of distinct flight levels allocated, symmetries among equivalent flight levels are dynamically broken, and the variable ordering is guided by the cliques found in the first static step. With an implementation using FaCiLe, our Functional Constraint Library, optimality is achieved for all flow sizes except the smallest one, while the corresponding number of flight levels could fit in the current airspace structure. However, many other constraints should be added to this very simplified model to obtain an operational route network, such that the conclusion is rather the validation of the concept of vertical separation of large flows. This graph coloring technique has also been tested on various benchmarks, featuring good results on reallife instances, which systematically appear to contain large cliques.
Stochastic local search algorithms for the graph set Tcolouring . . .
 APPROXIMATION ALGORITHMS AND METAHEURISTICS; COMPUTER AND INFORMATION SCIENCE SERIES
, 2005
"... The graph set Tcolouring 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 ..."
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Cited by 13 (3 self)
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The graph set Tcolouring 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.
An experimental investigation of iterated local search for coloring graphs
 Applications of Evolutionary Computing, volume 2270 of LNCS
, 2002
"... Abstract. Graph coloring is a well known problem from graph theory that, when attacldng it with local search algorithms, is typically treated as a series of constraint satisfaction problems: for a given number of colors k one has to find a feasible coloring: once such a coloring is found, the number ..."
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Cited by 10 (4 self)
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Abstract. Graph coloring is a well known problem from graph theory that, when attacldng it with local search algorithms, is typically treated as a series of constraint satisfaction problems: for a given number of colors k one has to find a feasible coloring: once such a coloring is found, the number of colors is decreased and the local search starts again. Here we explore the application of Iterated Local Search on the graph coloring problem. Iterated Local Search is a simple and powerful metaheuristic that has shown very good results for a variety of optimization problems. In our research we investigated several perturbation schemes and present computational results on a widely used set of benchmarks problems, a subset of those available from the DIMACS benchmark suite. Our results suggest that Iterated Local Search is particularly promising on hard, structured graphs.
Algorithms for Combinatorial Optimization in Real Time and their Automated Refinement by GeneticsBased Learning
 UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
, 1994
"... The goal of this research is to develop a systematic, integrated method of designing efficient search algorithms that solve optimization problems in real time. Search algorithms studied in this thesis comprise metacontrol and primitive search. The class of optimization problems addressed are called ..."
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Cited by 7 (1 self)
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The goal of this research is to develop a systematic, integrated method of designing efficient search algorithms that solve optimization problems in real time. Search algorithms studied in this thesis comprise metacontrol and primitive search. The class of optimization problems addressed are called combinatorial optimization problems, examples of which include many NPhard scheduling and planning problems, and problems in operations research and artificialintelligence applications. The problems we have addressed have a welldefined problem objective and a finite set of welldefined problem constraints. In this research, we use statespace trees as problem representations. The approach we have undertaken in designing efficient search algorithms is an engineering approach and consists of two phases: (a) designing generic search algorithms, and (b) improving by geneticsbased machine learning methods parametric heuristics used in the search algorithms designed. Our approach is a systematic method that integrates domain knowledge, search techniques, and automated learning techniques for designing better search algorithms. Knowledge captured in designing one search algorithm can be carried over for designing new ones.
Using an Incomplete Version of Dynamic Backtracking for Graph Colouring
, 1998
"... This paper investigates a simple, constructive search strategy: an incomplete version of Ginsberg's Dynamic Backtracking which records no information during search, and randomly selects variables for backtracking. Combining this with forward checking and dynamic variable ordering heuristics we ..."
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Cited by 5 (4 self)
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This paper investigates a simple, constructive search strategy: an incomplete version of Ginsberg's Dynamic Backtracking which records no information during search, and randomly selects variables for backtracking. Combining this with forward checking and dynamic variable ordering heuristics we obtain algorithms for constraint satisfaction problems, and by adding branchandbound we obtain optimization algorithms. Testing these on several classes of 1000vertex graph colouring problems we find colourings which are competitive with those of specialized algorithms: a 92colouring for a random graph (p=0.5), hidden kcolourings in equipartite graphs (p=0.5) for k 60 and improved results for geometric graphs. An interesting aspect is that most of our results were obtained using the precise opposite of the wellknown firstfail principle.