## Ants can solve Constraint Satisfaction Problems (2001)

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Venue: | IEEE Transactions on Evolutionary Computation |

Citations: | 24 - 9 self |

### BibTeX

@ARTICLE{Solnon01antscan,

author = {Christine Solnon},

title = {Ants can solve Constraint Satisfaction Problems},

journal = {IEEE Transactions on Evolutionary Computation},

year = {2001},

volume = {6},

pages = {347--357}

}

### Years of Citing Articles

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### Abstract

In this paper we describe a new incomplete approach for solving constraint satisfaction problems (CSPs) based on the ant colony optimization (ACO) metaheuristic. The idea is to use artificial ants to keep track of promising areas of the search space by laying trails of pheromone. This pheromone information is used to guide the search, as a heuristic for choosing values to be assigned to variables.

### Citations

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Citation Context ...s. A. The ant colony optimization metaheuristic ACO is a stochastic approach that has been proposed to solve dierent hard combinatorial optimization problems such as traveling salesman problems [6], [=-=10]-=-, [9], graph colouring problems [4], quadratic assignment problems [12], [20], and vehicle routing problems [1], [13]. The main idea of ACO is to model the problem as the search for a minimum cost pat... |

816 | Foundations of Constraint Satisfaction
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Citation Context ...ttern regognition, and machine vision. To solve CSPs, one may explore the search space in a systematic and complete way, until either a solution is found, or the problem is proven to have no solution =-=[31]-=-. In order to reduce the search space, this kind of complete approach is usually combined withsltering techniques that narrow the variables domains with respect to some partial consistencies. Complete... |

637 | Ant colony system: a cooperative learning approach to the traveling salesman problem
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Citation Context ...The ant colony optimization metaheuristic ACO is a stochastic approach that has been proposed to solve dierent hard combinatorial optimization problems such as traveling salesman problems [6], [10], [=-=9]-=-, graph colouring problems [4], quadratic assignment problems [12], [20], and vehicle routing problems [1], [13]. The main idea of ACO is to model the problem as the search for a minimum cost path in ... |

605 | Tabu Search
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Citation Context ... will concern the integration in Ant-Solver of other greedy local search heuristics (such as GSAT [25]), but also of local search procedures capable of escaping from local minima (such as tabu search =-=[15]-=- or random walk). Indeed, experimental comparisons of Ant-Solver with random walk reported in section VI show that these approaches have complementary features: random walk cansnd solutions to some pr... |

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Citation Context ...alues that belong to the generated solution. C. Phase-transitions When considering a class of combinatorial problems, rapid transitions in solvability may be observed as an order parameter is changed =-=[2]. The-=-se \phase-transitions" occur when evolving from instances that are under-constrained, and therefore solved rather easily, to instances that are over-constrained, whose inconsistency is thus prove... |

511 |
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Citation Context ...r of con icts, but which is not the best one. Hence, greedy local search has been combined with dierent meta-heuristics in order to help it escape from local optima. For example, Simulated Annealing [=-=25]-=- introduces a decreasing temperature parameter which enables local search to jump out of a local optimum by allowing moves towards worse assignments with a probability that both depends on the increas... |

422 | Partial constraint satisfaction
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Citation Context ...ll the constraints in C, that is, a complete assignment with zero cost. Most real-life CSPs are over-constrained, so that no solution exists. Hence, the CSP framework has been generalized to max-CSPs =-=[11-=-]. In this case, the goal is no longer tosnd a consistent solution, but tosnd a complete assignment that maximizes the number of satised constraints. Hence, a solution of a max-CSP is a complete assig... |

361 | B.: Noise strategies for improving local search
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Citation Context ... can be combined with dierent metaheuristics. We now compare our approach to the random walk MCH local search, denoted in the following by RWLS, which is one of the best-performing extensions of MCH [=-=24]-=-, [32]. At each step, the RWLS procedure either randomly changes the value of a variable, with probability p noise , or repairs a con icting variable using the min-con icts heuristic, with probability... |

317 | Ant algorithm for discrete optimization
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Citation Context ...using heuristics to guide the search towards the most-promising areas. In this paper, we describe an incomplete approach for solving CSPs based on the ant colony optimization (ACO) metaheuristic [8], =-=[7]. The-=- idea is to keep track of promising areas of the search space, by laying \pheromone" trails. This pheromone information is used to guide the search, as a heuristic for choosing values to be assig... |

296 | The ant colony optimization metaheuristic
- Dorigo, Caro
- 1999
(Show Context)
Citation Context ...lly, using heuristics to guide the search towards the most-promising areas. In this paper, we describe an incomplete approach for solving CSPs based on the ant colony optimization (ACO) metaheuristic =-=[8], [7]-=-. The idea is to keep track of promising areas of the search space, by laying \pheromone" trails. This pheromone information is used to guide the search, as a heuristic for choosing values to be ... |

285 | G.: Valued Constraint Satisfaction Problems: Hard and Easy Problems
- Schiex, Fargier, et al.
- 1995
(Show Context)
Citation Context ...l ants naturally solve optimization problems. In our context of constraint satisfaction, they aim at minimizing the number of violated constraints. This approach could also be extended to valued CSPs =-=[26]-=- in a rather straightforward way. B. Random binary CSPs Binary CSPs only have binary constraints, that is, each constraint involves two variables exactly. Binary CSPs may be generated at random. A cla... |

204 | Fitness distance correlation as a measure of problem difficultyforgeneticalgorithms.Proceedingsofthe6thInternationalConference on Genetic Algorithms
- Jones, Forrest
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Citation Context ...operty underlies the motivation of evolutionary approaches, that are of little interest in situations for which the correlation between solutionstness and the distance to optimal solutions is too low =-=[17]-=-, [21]. On the other hand, one also has to favor a large exploration of the search space, in order to discover new, and hopefully more successful, areas of the search space. Diversifying the search is... |

121 | Iterated local search
- Lourence, Martin, et al.
- 2002
(Show Context)
Citation Context ... to improve the assignment constructed. Various local search procedures may be used to improve assignments (see, e.g., [16] for an experimental comparison of some of them). However, as pointed out in =-=[18-=-], when choosing the local search procedure to use in a metaheuristic, such as evolutionary algorithms, iterated local search or ACO, one has tosnd a trade-o between computation time and solution qual... |

120 | MACS-VRPTW: A Multiple Ant Colony System for Vehicle Routing Problems with Time Windows
- Gambardella, Taillard, et al.
- 1999
(Show Context)
Citation Context ... hard combinatorial optimization problems such as traveling salesman problems [6], [10], [9], graph colouring problems [4], quadratic assignment problems [12], [20], and vehicle routing problems [1], =-=[13-=-]. The main idea of ACO is to model the problem as the search for a minimum cost path in a graph. Articial ants walk through this graph, looking for good paths. Each ant has a rather simple behaviour ... |

118 | The ant system applied to the quadratic assignment problem
- Maniezzo, Colorni
- 1999
(Show Context)
Citation Context ...that has been proposed to solve dierent hard combinatorial optimization problems such as traveling salesman problems [6], [10], [9], graph colouring problems [4], quadratic assignment problems [12], [=-=20-=-], and vehicle routing problems [1], [13]. The main idea of ACO is to model the problem as the search for a minimum cost path in a graph. Articial ants walk through this graph, looking for good paths.... |

116 | The constrainedness of search
- Gent, MacIntyre, et al.
- 1996
(Show Context)
Citation Context ...ven rather easily. Harder instances usually occur between these two kinds of \easy" instances, when approximately 50% of the instances are satisable. In order to predict the phase-transition regi=-=on, [14] in-=-troduces the notion of \constrainedness" of a class of problems, noted , and shows that when is close to 1, instances are critically constrained, and belong to the phase-transition region. For a... |

115 | Maniezzo V, Colorni A, Positive feedback as a search strategy - Dorigo - 1991 |

105 |
Ants can colour graphs
- Costa, Hertz
- 1997
(Show Context)
Citation Context ...taheuristic ACO is a stochastic approach that has been proposed to solve dierent hard combinatorial optimization problems such as traveling salesman problems [6], [10], [9], graph colouring problems [=-=4-=-], quadratic assignment problems [12], [20], and vehicle routing problems [1], [13]. The main idea of ACO is to model the problem as the search for a minimum cost path in a graph. Articial ants walk t... |

96 | GENET: A connectionist architecture for solving constraint satisfaction problems by iterative improvement
- Davenport, Tsang, et al.
- 1994
(Show Context)
Citation Context ... a population of good and representative complete assignments, and generate new candidates to be repaired by crossingover and/or mutating complete assignments from the population;sGuided Local Search =-=[30], [-=-31] escapes from local minima by increasing the weight of the violated constraints, in a eort to \ll up" the local minimum until local search escapes it; Iterated Local Search [32] iteratively pe... |

85 | An improved ant system algorithm for the vehicle routing problem
- Bullnheimer, Hartl, et al.
- 1999
(Show Context)
Citation Context ...erent hard combinatorial optimization problems such as traveling salesman problems [6], [10], [9], graph colouring problems [4], quadratic assignment problems [12], [20], and vehicle routing problems =-=[1-=-], [13]. The main idea of ACO is to model the problem as the search for a minimum cost path in a graph. Articial ants walk through this graph, looking for good paths. Each ant has a rather simple beha... |

79 | Ant colonies for the quadratic assignment problem
- Gambardella, Taillard, et al.
- 1999
(Show Context)
Citation Context ...roach that has been proposed to solve dierent hard combinatorial optimization problems such as traveling salesman problems [6], [10], [9], graph colouring problems [4], quadratic assignment problems [=-=12-=-], [20], and vehicle routing problems [1], [13]. The main idea of ACO is to model the problem as the search for a minimum cost path in a graph. Articial ants walk through this graph, looking for good ... |

79 | Max-min ant system
- Stutzle, Hoos
- 2000
(Show Context)
Citation Context ...ches such as local search. III. Description of Ant-Solver The overall algorithm for solving CSPs, called AntSolver, follows the classical ACO algorithmic scheme for static combinatorial problems [5], =-=[19]-=- and is sketched insgure 1. At each cycle of this algorithm, every ant constructs a complete assignment. Then, pheromone trails are updated with respect to the set of constructed assignments. The algo... |

77 | Easy problems are sometimes hard
- Gent, Walsh
- 1994
(Show Context)
Citation Context ... an edge between any pair of vertices corresponding to two dierent variables. More formally, the pheromone 1 However, exceptionally hard instances may also occur within the weakly constrained region [15]. procedure Ant-Solver let - (X; D;C) be the CSP to solve, - be pheromone trails, - ,s, and nbAnts be parameters. begin SetParameters(;s; ; nbAnts) InitializePheromoneTrails() repeat for k in ... |

75 |
Optimization, Learning and Natural Algorithms (in Italian
- Dorigo
- 1992
(Show Context)
Citation Context ...iables. A. The ant colony optimization metaheuristic ACO is a stochastic approach that has been proposed to solve dierent hard combinatorial optimization problems such as traveling salesman problems [=-=6]-=-, [10], [9], graph colouring problems [4], quadratic assignment problems [12], [20], and vehicle routing problems [1], [13]. The main idea of ACO is to model the problem as the search for a minimum co... |

58 | Fitness landscapes and memetic algorithm design
- Merz, Freisleben
- 1999
(Show Context)
Citation Context ... underlies the motivation of evolutionary approaches, that are of little interest in situations for which the correlation between solutionstness and the distance to optimal solutions is too low [17], =-=[21]-=-. On the other hand, one also has to favor a large exploration of the search space, in order to discover new, and hopefully more successful, areas of the search space. Diversifying the search is parti... |

50 | P.: Localizer: A modeling language for local search
- Michel, Hentenryck
- 1997
(Show Context)
Citation Context ...evaluated incrementally when constructing assignments, or when repairing them by local search. Actually, Ant-Solver could be integrated within a modelling language for local search, such as Localizer =-=[22]-=-. Such a language would allow one to design ecient and incremental procedures for local search more easily. A drawback of the genericity of Ant-Solver is that, on some particular problems, it may be l... |

45 |
A new method to solve hard satis problems
- Selman, Levesque, et al.
- 1992
(Show Context)
Citation Context ... experiments reported in this paper, we used the min-con icts heuristic for local search. Further work will concern the integration in Ant-Solver of other greedy local search heuristics (such as GSAT =-=[25]-=-), but also of local search procedures capable of escaping from local minima (such as tabu search [15] or random walk). Indeed, experimental comparisons of Ant-Solver with random walk reported in sect... |

45 |
A filtering algorithm for global sequencing constraints
- Régin, Puget
- 1997
(Show Context)
Citation Context ...ining unassigned variables. Hence, there are very few heuristics for guiding this choice and these heuristics usually are problem-dependent so that they cannot be applied to general CSPs (e.g., [20], =-=[21]-=- for the car-sequencing problem). The main contribution of Ant-Solver is to propose a general heuristic for choosing values, based on the ACO meta-heuristic. More precisely, the choice of a value for ... |

44 | Local search and the number of solutions
- Clark, Frank, et al.
- 1996
(Show Context)
Citation Context ...ss" of a class of problems, noted , and shows that when is close to 1, instances are critically constrained, and belong to the phase-transition region. For a class of random binary CSPss1 ; p 2 =-=>, [-=-3] denes this constrainedness by = n 1 2 p 1 log m ( 1 1 p2 ). One might think that phase-transitions only concern complete approaches, as they are usually associated with transitions from solvable t... |

43 |
Guided Local Search for Combinatorial Optimisation Problems
- Voudouris
- 1997
(Show Context)
Citation Context ...ulation of good and representative complete assignments, and generate new candidates to be repaired by crossingover and/or mutating complete assignments from the population;sGuided Local Search [30], =-=[31] es-=-capes from local minima by increasing the weight of the violated constraints, in a eort to \ll up" the local minimum until local search escapes it; Iterated Local Search [32] iteratively perturba... |

34 | Tabu search for maximal constraint satisfaction problems
- Galinier, Hao
- 1997
(Show Context)
Citation Context ...bility problems (SAT), but it can be extended to solve CSPs in a very straightforward way. SUBMISSION TO IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, OCTOBER 2001 106 temperature; Tabu Search [26], =-=[27-=-] prevents local search from cycling through a small set of good but suboptimal points by keeping track of a buer of forbidden moves in a TABU list; Genetic Algorithms [28], [29] maintain a population... |

32 | Modelling a permutation problem
- Smith
- 2000
(Show Context)
Citation Context ...iables to be assigned with a permutation of n known values). Many CSPs involve such global permutation constraints, e.g., the n-queens, the all-interval series problem, and the car-sequencing problem =-=[28]-=-. For these particular constraints, we have introduced a dedicated graph that is much smaller than the construction graph used in AntSolver, and that takes permutation constraints into account in an a... |

30 | Random constraint satisfaction: theory meets practice
- MacIntyre, Prosser, et al.
- 1998
(Show Context)
Citation Context ...p 2 determines the number of incompatible pairs of values for each constraint). Experiments reported in this paper were obtained with random binary CSPs generated according to model A as described in =-=[19]-=-, that is, for each pair of distinct variables, we add a constraint between them with probability p 1 , and then for each constraint we rule out a pair of values for the two constrained variables with... |

30 | A Tutorial on Constraint Programming
- Smith
- 1995
(Show Context)
Citation Context ...th some other related approaches and discusses further works. II. Background A. CSPs We now brie y recall some denitions and terminology about CSPs. More details can be found, for example, in [3] or [=-=-=-10]. A CSP is dened by a triple (X; D;C) such that X = fX 1 ; X 2 ; :::; Xn g is asnite set of n variables, D is a function which maps every variable X i 2 X to its domain D(X i ), i.e., the set of ... |

28 | Heuristic methods for over-constrained constraint satisfaction problems
- Wallace, Freuder
- 1106
(Show Context)
Citation Context ...e combined with dierent metaheuristics. We now compare our approach to the random walk MCH local search, denoted in the following by RWLS, which is one of the best-performing extensions of MCH [24], [=-=32]-=-. At each step, the RWLS procedure either randomly changes the value of a variable, with probability p noise , or repairs a con icting variable using the min-con icts heuristic, with probability 1 p n... |

20 |
Minimizing con A heuristic repair method for constraint satisfaction and scheduling problems
- Minton, Johnston, et al.
- 1992
(Show Context)
Citation Context ... , although it is more emphasized for the hardest instances. IV. Boosting Ant-Solver with local search Local search has shown to be eective to solve large CSPs, e.g., the \million queens problem"=-= in [23]-=-. The basic idea is to construct a complete assignment and then gradually and iteratively repair it by changing some variablevalue assignments. Local search may be combined with the ACO metaheuristic ... |

19 |
The CHIP system and its applications
- Simonis
- 1995
(Show Context)
Citation Context ...gnition or machine vision. To solve CSPs, one can explore the search space in a systematic and complete way, until either a solution is found, or the problem is proved to have no solution (e.g., [1], =-=[2-=-]). In order to reduce the search space, this kind of complete approach is usually combined with propagation /ltering techniques that narrow variable domains with respect to some partial consistencies... |

17 |
adding more constraints makes a problem easier for hill-climbing algorithms: Analysing landscapes of csp’s
- Why
- 1997
(Show Context)
Citation Context ...rom the phase-transition region. However, it gives very poor results for instances within the phase-transition region, as the search space landscape of these instances contains many more local optima =-=[33-=-], [3]. To help it escape from local optima, greedy local search can be combined with dierent metaheuristics. We now compare our approach to the random walk MCH local search, denoted in the following ... |

15 | Empirical studies of heuristic local search for constraint solving
- Hao, Dorne
- 1996
(Show Context)
Citation Context ...signment, and before updating pheromone trails, we apply a local search procedure to improve the assignment constructed. Various local search procedures may be used to improve assignments (see, e.g., =-=[16]-=- for an experimental comparison of some of them). However, as pointed out in [18], when choosing the local search procedure to use in a metaheuristic, such as evolutionary algorithms, iterated local s... |

11 | 2000) Solving permutation constraint satisfaction problems with artificial ants
- Solnon
(Show Context)
Citation Context ... MAY 2002 10 equivalent set of simpler constraints, and then use AntSolver. However, one usually obtains better results when using specialized algorithms for these global constraints. For example, in =-=[29]-=- we have described an ACO algorithm for handling permutation constraints (that constrain a set of n variables to be assigned with a permutation of n known values). Many CSPs involve such global permut... |

10 | A comparison of complete and incomplete algorithms in the easy and hard regions
- Davenport
- 1995
(Show Context)
Citation Context ...ern complete approaches, as they are usually associated with transitions from solvable to unsolvable instances, and incomplete approaches cannot detect unsolvability. However, dierent studies (e.g., [=-=5]-=-, [3]) have shown that very similar phase-transition phenomena may also be observed with incomplete approaches such as local search. III. Description of Ant-Solver The proposed algorithm for solving C... |

8 | A genetic local search algorithm for random binary constraint satisfaction problems
- Marchiori, Steenbeek
- 2000
(Show Context)
Citation Context ...perature; Tabu Search [26], [27] prevents local search from cycling through a small set of good but suboptimal points by keeping track of a buer of forbidden moves in a TABU list; Genetic Algorithms [=-=28]-=-, [29] maintain a population of good and representative complete assignments, and generate new candidates to be repaired by crossingover and/or mutating complete assignments from the population;sGuide... |

7 |
How adding more constraints makes a problem easier for hill-climbing algorithms: Analyzing landscapes of CSPs
- Yokoo
- 1997
(Show Context)
Citation Context ...ion, problems are hardly constrained and only have few solutions, but they also have much less local optima so that local search can more easily reach a solution without being trapped in local optima =-=[33]-=-, [17]. However, search space landscapes of problems within the phase transition region contain more local minima so that local search is more often trapped in these local minima and, even when using ... |

6 |
Ilog Solver 4.4, User’s Manual
- ILOG
- 1999
(Show Context)
Citation Context ... regognition or machine vision. To solve CSPs, one can explore the search space in a systematic and complete way, until either a solution is found, or the problem is proved to have no solution (e.g., =-=[1-=-], [2]). In order to reduce the search space, this kind of complete approach is usually combined with propagation /ltering techniques that narrow variable domains with respect to some partial consiste... |

6 |
Tackling car sequencing problems using a genetic algorithm
- Warwick, Tsang
- 1995
(Show Context)
Citation Context ...re; Tabu Search [26], [27] prevents local search from cycling through a small set of good but suboptimal points by keeping track of a buer of forbidden moves in a TABU list; Genetic Algorithms [28], [=-=29]-=- maintain a population of good and representative complete assignments, and generate new candidates to be repaired by crossingover and/or mutating complete assignments from the population;sGuided Loca... |

2 |
Succeed- or fail- A case study in variable and value ordering heuristics
- Smith
- 1996
(Show Context)
Citation Context ...iables are assigned is rather important, as the satisfaction of a constraint can only be checked once all its variables have been assigned. Variable-ordering heuristics have been studied widely [31], =-=[27-=-]. Some commonly used variable-orderings are: most-constraining-rst ordering, which selects an unassigned variable that is connected (by a constraint) to the largest number of unassigned variables; mo... |