Maximum Constraint Satisfaction Research Articles

Near-optimal algorithms for maximum constraint satisfaction problems

In this article, we present two approximation algorithms for the maximum constraint satisfaction problem with k variables in each constraint (MAX k -CSP). Given a (1 − ε) satisfiable 2CSP our first algorithm finds an assignment of variables satisfying a 1 − O (√ε) fraction of all constraints. The best previously known result, due to Zwick, was 1 − O (ε 1/3 ). The second algorithm finds a ck /2 k approximation for the MAX k -CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which had an approximation guarantee of Ω( k /(2 k log k )). Both results are optimal assuming the unique games conjecture and are based on rounding natural semidefinite programming relaxations. We also believe that our algorithms and their analysis are simpler than those previously known.

Hard constraint satisfaction problems have hard gaps at location 1

An instance of the maximum constraint satisfaction problem ( Max CSP) is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max k - SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint language) affect the complexity and approximability of Max CSP. The PCP theorem is equivalent to the existence of a constraint language for which Max CSP has a hard gap at location 1; i.e. it is NP-hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints can be satisfied) is currently known to be NP-hard, have a certain algebraic property. We prove that any constraint language with this algebraic property makes Max CSP have a hard gap at location 1 which, in particular, implies that such problems cannot have a PTAS unless P = NP . We then apply this result to Max CSP restricted to a single constraint type; this class of problems contains, for instance, Max Cut and Max DiCut. Assuming P ≠ NP, we show that such problems do not admit PTAS except in some trivial cases. Our results hold even if the number of occurrences of each variable is bounded by a constant. Finally, we give some applications of our results.

The approximability of MAX CSP with fixed-value constraints

In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. In this article, we show that any MAX CSP problem with a finite set of allowed constraint types, which includes all fixed-value constraints (i.e., constraints of the form x = a ), is either solvable exactly in polynomial time or else is APX-complete, even if the number of occurrences of variables in instances is bounded. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description relies on the well-known algebraic combinatorial property of supermodularity.

Maximizing Supermodular Functions on Product Lattices, with Application to Maximum Constraint Satisfaction

Recently, a strong link has been discovered between supermodularity on lattices and tractability of optimization problems known as maximum constraint satisfaction problems. This paper strengthens this link. We study the problem of maximizing a supermodular function which is defined on a product of $n$ copies of a fixed finite lattice and given by an oracle. We exhibit a large class of finite lattices for which this problem can be solved in oracle-polynomial time in $n$. We also obtain new large classes of tractable maximum constraint satisfaction problems.

Maximum H-colourable subdigraphs and constraint optimization with arbitrary weights

In the maximum constraint satisfaction problem ( Max CSP), one is given a finite collection of positive-weight constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so that the total weight of satisfied constraints is maximized. We consider this problem and its variant Max AW CSP where the weights are allowed to be both positive and negative, and study how the complexity of the problems depends on the allowed constraint types. We prove that Max AW CSP over an arbitrary finite domain exhibits a dichotomy: it is either polynomial-time solvable or NP-hard. Our proof builds on two results that may be of independent interest: one is that the problem of finding a maximum H-colourable subdigraph in a given digraph is either NP-hard or trivial depending on H, and the other a dichotomy result for Max CSP with a single allowed constraint type.

A family of distributed double guided genetic algorithm for Max_CSPs

This paper presents, studies and betters distributed Guided Genetic Algorithm (DGGA) dealing with Maximal Constraint Satisfaction Problems. This algorithm consists of agents dynamically created and cooperating in order to satisfy the maximal number of constraints. Each agent performs its own GA, gu ided by both the template concept and the Min-conflict-heuristic, on a sub-population composed of chromosomes violating the same number of constraints. D2G2A is a new multi-agent approach, which in addition to DGGA will be enhanced by a new parameter called guidance operator. The latter allows not only diversification but also an escaping from local optima. D2G2A is improved in the second part. This improvement is based on the NEO-DARWINISM theory and on the laws of nature. In fact, the new algorithm will let the species agent able to count its cross-over probability and its mutation probability. This approach is called D3G2A. In this paper, newer algorithms and their global dynamics are furnished, and experimental results are provided.

A family of distributed double guided genetic algorithm for Max_CSPs

This paper presents, studies and betters distributed Guided Genetic Algorithm (DGGA) dealing with Maximal Constraint Satisfaction Problems. This algorithm consists of agents dynamically created and...

The Approximability of Three-valued MAX CSP

In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) MAX CSP with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APX-complete (and hence can have no polynomial-time approximation scheme unless P=NP). It has been an open problem for several years whether this result can be extended to non-Boolean MAX CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for MAX CSP over a three-element domain. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known algebraic combinatorial property of supermodularity. We also show that every hard three-valued MAX CSP contains, in a certain specified sense, one of the two basic hard MAX CSPs which are the Maximum k-Colorable Subgraph problems for k=2,3.

Supermodular functions and the complexity of MAX CSP

In this paper we study the complexity of the maximum constraint satisfaction problem (MAX CSP) over an arbitrary finite domain. An instance of MAX CSP consists of a set of variables and a collectio...

A Genetic Algorithm for Searching Spatial Configurations

Searching spatial configurations is a particular case of maximal constraint satisfaction problems, where constraints expressed by spatial and nonspatial properties guide the search process. In the spatial domain, binary spatial relations are typically used for specifying constraints while searching spatial configurations. Searching configurations is particularly intractable when configurations are derived from a combination of objects, which involves a hard combinatorial problem. This paper presents a genetic algorithm (GA) that combines a direct and an indirect approach to treating binary constraints in genetic operators. A new genetic operator combines randomness and heuristics for guiding the reproduction of new individuals in a population. Individuals are composed of spatial objects whose relationships are indexed by a content measure. This paper describes the GA and presents experimental results that compare the genetic versus a deterministic and a local-search algorithm. These experiments show the convenience of using a GA when the complexity of the queries and databases do no guarantee the tractability of a deterministic strategy.

Supermodular functions and the complexity of M AX CSP

In this paper we study the complexity of the maximum constraint satisfaction problem (M AX CSP) over an arbitrary finite domain. An instance of M AX CSP consists of a set of variables and a collection of constraints which are applied to certain specified subsets of these variables; the goal is to find values for the variables which maximize the number of simultaneously satisfied constraints. Using the theory of sub- and supermodular functions on finite lattice-ordered sets, we obtain the first examples of general families of efficiently solvable cases of M AX CSP for arbitrary finite domains. In addition, we provide the first dichotomy result for a special class of non-Boolean M AX CSP, by considering binary constraints given by supermodular functions on a totally ordered set. Finally, we show that the equality constraint over a non-Boolean domain is non-supermodular, and, when combined with some simple unary constraints, gives rise to cases of M AX CSP which are hard even to approximate.

Supermodularity on chains and complexity of maximum constraint satisfaction

In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity.

Nurse Scheduling System based on Dynamic Weighted Maximal Constraint Satisfaction Problem-Solution Stability based on the Provisional Constraint-

Nurse Scheduling System based on Dynamic Weighted Maximal Constraint Satisfaction Problem-Solution Stability based on the Provisional Constraint-

Polynomial time approximation schemes for dense instances of minimum constraint satisfaction

AbstractIt is known that large fragments of the class of dense Minimum Constraint Satisfaction (MIN‐CSP) problems do not have polynomial time approximation schemes (PTASs) contrary to their Maximum Constraint Satisfaction analogs. In this paper we prove, somewhat surprisingly, that the minimum satisfaction of dense instances of kSAT‐formulas, and linear equations mod 2, Ek‐LIN2, do have PTASs for any k. The MIN‐Ek‐LIN2 problems are equivalent to the k‐ary versions of the Nearest Codeword problem, the problem which is known to be exceedingly hard to approximate on general instances. The method of solution of the above problems depends on the development of a new density sampling technique for k‐uniform hypergraphs which could be of independent interest. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 73–91, 2003

Speed Up of Distributed Maximal Constraint Satisfaction Algorithm

Speed Up of Distributed Maximal Constraint Satisfaction Algorithm

Reactive local search techniques for the maximum k-conjunctive constraint satisfaction problem ( MAX-k-CCSP)

In this paper the performance of the Hamming-based Reactive Tabu Search algorithm (H-RTS) previously proposed for the Maximum Satisfiability problem is studied for the different Maximum k-Conjunctive Constraint Satisfaction problem. In addition, the use of non-oblivious functions recently proposed in the framework of approximation algorithms is investigated. In particular, two relevant special cases of the Maximum k-Conjunctive Constraint Satisfaction problem are considered: Maximum Directed Cut and Maximum Independent Set in cubic graphs. The preliminary diversification-bias analysis of the basic components shows a remarkable difference between the two problems, and the derived predictions are then validated by extensive experiments with the complete H-RTS algorithm. The performance of H-RTS is compared with that of Simulated Annealing and simple Repeated Local Search.

A simulated annealing artificial neural network implementation of the n‐queens problem

A Harmony Theory artificial neural network implementation of the n-queens problem is presented in this piece of research. The problem is encoded in the two layers of the artificial neural network in such a manner that the inherent constraints of the problem are made directly available. Subsequently, during the simulated annealing procedure of Harmony Theory, maximal constraint satisfaction is accomplished in parallel and an optimal solution of the n-queens problem is produced. This solution indicates the appropriate locations of the greatest possible number of queens that can be placed on the n × n chessboard in a valid configuration, i.e., so that no queen threatens or is threatened by another queen. The proposed parallel implementation of the n-queens problem, combined with the application of the simulated annealing procedure, offers an interesting alternative to existing techniques (e.g., search, constraint propagation) in terms of optimality as well as computational and time efficiency. © 1997 John Wiley & Sons, Inc.

A simulated annealing artificial neural network implementation of the n-queens problem

A Harmony Theory artificial neural network implementation of the n-queens problem is presented in this piece of research. The problem is encoded in the two layers of the artificial neural network in such a manner that the inherent constraints of the problem are made directly available. Subsequently, during the simulated annealing procedure of Harmony Theory, maximal constraint satisfaction is accomplished in parallel and an optimal solution of the n-queens problem is produced. This solution indicates the appropriate locations of the greatest possible number of queens that can be placed on the n × n chessboard in a valid configuration, i.e., so that no queen threatens or is threatened by another queen. The proposed parallel implementation of the n-queens problem, combined with the application of the simulated annealing procedure, offers an interesting alternative to existing techniques (e.g., search, constraint propagation) in terms of optimality as well as computational and time efficiency. © 1997 John Wiley & Sons, Inc.