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26
The hardest constraint problems: A double phase transition
 Artif. Intell
, 1994
"... The distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors. The first, previously recognized, is a peak in the median search cost near the connectivity at which half the graphs have solutions. This region contains a high pr ..."
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Cited by 95 (2 self)
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The distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors. The first, previously recognized, is a peak in the median search cost near the connectivity at which half the graphs have solutions. This region contains a high proportion of relatively hard problem instances. However, the hardest instances are in fact concentrated at a second, lower, transition point. Near this point, most problems are quite easy, but there are also a few very hard cases. This region of exceptionally hard problems corresponds to the transition between polynomial and exponential scaling of the average search cost, whose location we also estimate theoretically. These behaviors also appear to arise in other constraint problems. This work also shows the limitations of simple measures of the cost distribution, such as mean or median, for identifying outlying cases. 1
"Squeaky Wheel" Optimization
, 1999
"... 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 fun ..."
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Cited by 84 (3 self)
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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 reprioritization 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 ...
Statistical mechanics of combinatorial search
 In Proc. of the Workshop on Physics and Computation (PhysComp94
, 1994
"... The statistical mechanics of combinatorial search problems is described using the example of the wellknown NPcomplete graph coloring problem. We focus on a recently identified phase transition from under to overconstrained problems, near which are concentrated many hard to solve search problems. ..."
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Cited by 21 (5 self)
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The statistical mechanics of combinatorial search problems is described using the example of the wellknown NPcomplete graph coloring problem. We focus on a recently identified phase transition from under to overconstrained problems, near which are concentrated many hard to solve search problems. Thus, a readily computed measure of problem structure predicts the difficulty of solving the problem, on average. However, this prediction is associated with a large variance and depends on the somewhat arbitrary choice of the problem ensemble. Thus these results are of limited direct use for individual instances. To help address this limitation, additional parameters, describing problem structure as well as heuristic effectiveness, are introduced. This also highlights the distinction between the statistical mechanics of combinatorial search problems, with their exponentially large search spaces, and physical systems, whose interactions are often governed by a simple euclidean metric. Chapter 1
Phase Transitions from Real Computational Problems
 In Proceedings of the 8th International Symposium on Artificial Intelligence
, 1995
"... We examine phase transitions in problems derived from real computational problems using a wide variety of algorithms. These phase transitions resemble those observed with randomly generated problems. Real problems do, however, contain new features (e.g. large scale structures rare in random problems ..."
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Cited by 20 (8 self)
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We examine phase transitions in problems derived from real computational problems using a wide variety of algorithms. These phase transitions resemble those observed with randomly generated problems. Real problems do, however, contain new features (e.g. large scale structures rare in random problems) which can make them significantly harder than random problems. Our results suggest a new methodology for benchmarking algorithms. In addition, they help to identify the location of the really hard real problems. 1 Introduction A conventional method for comparing the performance of algorithms is to use benchmark problems. Since the supply of benchmark problems is usually limited, we may be unable to perform a statistically significant comparison, or to determine accurately how performance depends on problem size and problem difficulty. An alternative approach is to use random problems which are cheap to generate at different problem sizes. Unfortunately random problems are typically easy...
An Unconstrained Quadratic Binary Programming Approach to the Vertex Coloring Problem
, 2003
"... The vertex coloring problem has been the subject of extensive research for many years. Driven by application potential as well as computational challenge, a variety of methods have been proposed for this difficult class of problems. Recent successes in the use of the unconstrained quadratic programm ..."
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Cited by 17 (7 self)
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The vertex coloring problem has been the subject of extensive research for many years. Driven by application potential as well as computational challenge, a variety of methods have been proposed for this difficult class of problems. Recent successes in the use of the unconstrained quadratic programming (UQP) model as a unified framework for modeling and solving combinatorial optimization problems have motivated a new approach to the vertex coloring problem. In this paper we present a UQP approach to this problem and illustrate its attractiveness with preliminary computational experience.
A GRASP for Coloring Sparse Graphs
, 1999
"... We first present a literature review of heuristics and metaheuristics developed for the problem of coloring graphs. We then present a Greedy Randomized Adaptive Search Procedure (GRASP) for coloring sparse graphs. The procedure is tested on graphs of known chromatic number, as well as random graphs ..."
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Cited by 16 (0 self)
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We first present a literature review of heuristics and metaheuristics developed for the problem of coloring graphs. We then present a Greedy Randomized Adaptive Search Procedure (GRASP) for coloring sparse graphs. The procedure is tested on graphs of known chromatic number, as well as random graphs with edge probability 0.1 having from 50 to 500 vertices. Empirical results indicate that the proposed GRASP implementation compares favorably to classical heuristics and implementations of simulated annealing and tabu search. GRASP is also found to be competitive with a genetic algorithm that is considered one of the best currently available for graph coloring.
Coloration Neighbourhood Search With Forward Checking
 Annals of Mathematics and Artificial Intelligence
, 2002
"... 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. Nei ..."
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Cited by 15 (10 self)
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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.
A Search Space “Cartography” for Guiding Graph Coloring Heuristics
 International Conferences
"... 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 spac ..."
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Cited by 11 (7 self)
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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 (TSDiv) uses a learning process to guide the basic TS process toward asyetunvisited spheres. The second algorithm (TSInt) makes deep investigations within a bounded region by organizing it as a treelike structure of connected spheres. We experimentally demonstrate that if such a region contains a global optimum, TSInt 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 bestknown solutions ever reported in the literature (e.g. for dsjc1000.9). Key words: graph coloring, local optima distribution, search by learning. 1
Constrained Bandwidth Multicoloration Neighbourhoods
"... A combination of local search and constraint propagation called FCNS has previously been described for pure graph colouring. FCNS is a hybrid of the DSATUR backtracker and the IMPASSE local search algorithm, restricting coloration neighbourhoods by propagation, and dynamically ordering vertices b ..."
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Cited by 6 (1 self)
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A combination of local search and constraint propagation called FCNS has previously been described for pure graph colouring. FCNS is a hybrid of the DSATUR backtracker and the IMPASSE local search algorithm, restricting coloration neighbourhoods by propagation, and dynamically ordering vertices by the Brelaz heuristic. We extend it to bandwidth colouring and present results on a set of geometric graphs. A related algorithm called Saturn has previously been described for pure 0/1 integer linear programs. We model bandwidth multicolouring as an ILP and present Saturn results on the same graphs. An assessment of performance is postponed until other results are presented at the symposium.
Exploiting relaxation in local search
 First International Workshop on Local Search Techniques in Constraint Satisfaction
, 2004
"... Abstract. Branchandbound uses relaxation to prune search trees but sometimes scales poorly to large problems. Conversely, local search often scales well but may be unable to find optimal solutions, perhaps because it does not exploit relaxation. Both phenomena occur in the construction of lowauto ..."
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Cited by 6 (0 self)
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Abstract. Branchandbound uses relaxation to prune search trees but sometimes scales poorly to large problems. Conversely, local search often scales well but may be unable to find optimal solutions, perhaps because it does not exploit relaxation. Both phenomena occur in the construction of lowautocorrelation binary sequences, a problem arising in communication engineering. This paper proposes a hybrid approach to optimization: using relaxation to prune local search spaces. An implementation gives competitive results, showing the feasibility of the approach. 1