Weighted Constraint Satisfaction Problems with Min-Max Quantifiers

Many combinatorial problems deal with preferences and violations, the goal of which is to find solutions with the minimum cost. Weighted constraint satisfaction is a framework for modeling such problems, which consists of a set of cost functions to measure the degree of violation or preferences of different combinations of variable assignments. Typical solution methods for weighted constraint satisfaction problems (WCSPs) are based on branch-and-bound search, which are made practical through the use of powerful consistency techniques such as AC*, FDAC*, EDAC* to deduce hidden cost information and value pruning during search. These techniques, however, are designed to be efficient only on binary and ternary cost functions which are represented in table form. In tackling many real-life problems, high arity (or global) cost functions are required. We investigate efficient representation scheme and algorithms to bring the benefits of the consistency techniques to also high arity cost functions, which are often derived from hard global constraints from classical constraint satisfaction. The literature suggests some global cost functions can be represented as flow networks, and the minimum cost flow algorithm can be used to compute the minimum costs of such networks in polynomial time. We show that naive adoption of this flow-based algorithmic method for global cost functions can result in a stronger form of null-inverse consistency. We further show how the method can be modified to handle cost projections and extensions to maintain generalized versions of AC* and FDAC* for cost functions with more than two variables. Similar generalization for the stronger EDAC* is less straightforward. We reveal the oscillation problem when enforcing EDAC* on cost functions sharing more than one variable. To avoid oscillation, we propose a weak version of EDAC* and generalize it to weak EDGAC* for non-binary cost functions. Using various benchmarks involving the soft variants of hard global constraints ALLDIFFERENT, GCC, SAME, and REGULAR, empirical results demonstrate that our proposal gives improvements of up to an order of magnitude when compared with the traditional constraint optimization approach, both in terms of time and pruning.

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Solving Weighted Constraint Satisfaction Problems with Memetic/Exact Hybrid Algorithms

Journal of Artificial Intelligence Research ◽

10.1613/jair.2770 ◽

2009 ◽

Vol 35 ◽

pp. 533-555 ◽

Cited By ~ 11

Author(s):

J. E. Gallardo ◽

C. Cotta ◽

A. J. Fernández

Keyword(s):

Constraint Satisfaction ◽

Large Scale ◽

Constraint Satisfaction Problem ◽

Heuristic Method ◽

Optimal Solution ◽

Constraint Satisfaction Problems ◽

Still Life ◽

Level Model ◽

Weighted Constraint

A weighted constraint satisfaction problem (WCSP) is a constraint satisfaction problem in which preferences among solutions can be expressed. Bucket elimination is a complete technique commonly used to solve this kind of constraint satisfaction problem. When the memory required to apply bucket elimination is too high, a heuristic method based on it (denominated mini-buckets) can be used to calculate bounds for the optimal solution. Nevertheless, the curse of dimensionality makes these techniques impractical on large scale problems. In response to this situation, we present a memetic algorithm for WCSPs in which bucket elimination is used as a mechanism for recombining solutions, providing the best possible child from the parental set. Subsequently, a multi-level model in which this exact/metaheuristic hybrid is further hybridized with branch-and-bound techniques and mini-buckets is studied. As a case study, we have applied these algorithms to the resolution of the maximum density still life problem, a hard constraint optimization problem based on Conway's game of life. The resulting algorithm consistently finds optimal patterns for up to date solved instances in less time than current approaches. Moreover, it is shown that this proposal provides new best known solutions for very large instances.

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A Framework for Hybrid Tractability Results in Boolean Weighted Constraint Satisfaction Problems

Lecture Notes in Computer Science - Principles and Practice of Constraint Programming ◽

10.1007/978-3-540-85958-1_19 ◽

2008 ◽

pp. 282-297 ◽

Cited By ~ 8

Author(s):

T. K. Satish Kumar

Keyword(s):

Constraint Satisfaction ◽

Constraint Satisfaction Problems ◽

Weighted Constraint

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Parameterized Heuristics for Incomplete Weighted CSPs

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v33i01.33019898 ◽

2019 ◽

Vol 33 ◽

pp. 9898-9899

Author(s):

Atena M. Tabakhi

Keyword(s):

Error Bound ◽

Constraint Satisfaction ◽

A Priori ◽

Constraint Satisfaction Problems ◽

Optimal Solutions ◽

Solution Quality ◽

Trade Off ◽

Suboptimal Solutions ◽

Weighted Constraint

The key assumption in Weighted Constraint Satisfaction Problems (WCSPs) is that all constraints are specified a priori. This assumption does not hold in some applications that involve users preferences. Incomplete WCSPs (IWCSPs) extend WCSPs by allowing some constraints to be partially specified. Unfortunately, existing IWCSP approaches either guarantee to return optimal solutions or not provide any quality guarantees on solutions found. To bridge the two extremes, we propose a number of parameterized heuristics that allow users to find boundedly-suboptimal solutions, where the error bound depends on user-defined parameters. These heuristics thus allow users to trade off solution quality for fewer elicited preferences and faster computation times.

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Solving Weighted Constraint Satisfaction Problems Using a new Self-Adaptive Discrete Firefly Algorithm

2019 IEEE International Conference on Systems, Man and Cybernetics (SMC) ◽

10.1109/smc.2019.8914624 ◽

2019 ◽

Cited By ~ 1

Author(s):

Mahdi Bidar ◽

Malek Mouhoub

Keyword(s):

Constraint Satisfaction ◽

Firefly Algorithm ◽

Constraint Satisfaction Problems ◽

Discrete Firefly Algorithm ◽

Weighted Constraint ◽

Self Adaptive

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Parameterized Heuristics for Incomplete Weighted CSPs

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v33i01.330110045 ◽

2019 ◽

Vol 33 ◽

pp. 10045-10046

Author(s):

Atena M. Tabakhi

Keyword(s):

Error Bound ◽

Constraint Satisfaction ◽

A Priori ◽

Constraint Satisfaction Problems ◽

Optimal Solutions ◽

Solution Quality ◽

Trade Off ◽

Suboptimal Solutions ◽

Weighted Constraint

The key assumption in Weighted Constraint Satisfaction Problems (WCSPs) is that all constraints are specified a priori. This assumption does not hold in some applications that involve users preferences. Incomplete WCSPs (IWCSPs) extend WCSPs by allowing some constraints to be partially specified. Unfortunately, existing IWCSP approaches either guarantee to return optimal solutions or not provide any quality guarantees on solutions found. To bridge the two extremes, we propose a number of parameterized heuristics that allow users to find boundedly-suboptimal solutions, where the error bound depends on user-defined parameters. These heuristics thus allow users to trade off solution quality for fewer elicited preferences and faster computation times.

Download Full-text