Arc Consistency Algorithm Research Articles

Using Constraint Propagation to Bound Linear Programs

We present an approach to compute bounds on the optimal value of linear programs based on constraint propagation. Given a feasible dual solution, we apply constraint propagation to the complementary slackness conditions and, if propagation succeeds to prove these conditions infeasible, the infeasibility certificate (in the sense of Farkas’ lemma) is reconstructed from the propagation history. This certificate is a dual-improving direction, which allows us to improve the bound. As constraint propagation need not always detect infeasibility of a linear inequality system, the method is not guaranteed to converge to a global solution of the linear program but only to an upper bound, whose tightness depends on the structure of the program and the used propagation method. The approach is suited for large sparse linear programs (such as LP relaxations of combinatorial optimization problems), for which the classical LP algorithms may be infeasible, if only for their super-linear space complexity. The approach can be seen as a generalization of the Virtual Arc Consistency (VAC) algorithm to bound the LP relaxation of the Weighted CSP (WCSP). We newly apply it to the LP relaxation of the Weighted Max-SAT problem, experimentally showing that the obtained bounds are often not far from optima of the relaxation and proving that they are exact for known tractable subclasses of Weighted Max-SAT.

A mark-based hierarchical asynchronous temporal constraints reasoning algorithm for multi-agent mission planning

The establishment of a lunar base requires the collaborative cooperation of multiple lunar rovers. The efficiency of rovers’ mission planning can be increased by quickly determining whether the numerous temporal constraints between the activities of rovers are consistent. This research focuses on constructing a new and efficient algorithm to solve the complex temporal constraint problem in mission planning for lunar rovers. A mark-based centralized arc-consistency algorithm is proposed to filter unnecessary constraint checks depending on whether the mark of each variable changes. It improves computational efficiency and we prove the rationality of the algorithm theoretically. Then, based on the algorithm mentioned above, a mark-based hierarchical asynchronous algorithm is proposed to reduce the iterations of temporal constraints reasoning through asynchronous computation. At the same time, each rover tightens its local variables’ domains to a minimum before interacting with other rovers, which reduces the impact of locally shared variables on the outside. This avoids the problem of reduced computational efficiency caused by the waiting time of asynchronous computing. Finally, we conduct comprehensive experiments on different types of benchmark datasets, and the results show that the mark-based hierarchical asynchronous algorithm can improve efficiency by up to 47.64 % and reduce the number of constraint checks by up to 89.26 % compared to existing algorithms, which also benefits to reduce the number of communication messages and iterations.

Efficient parallel graph trimming by arc-consistency

Given a large data graph, trimming techniques can reduce the search space by removing vertices without outgoing edges. One application is to speed up the parallel decomposition of graphs into strongly connected components (SCC decomposition), which is a fundamental step for analyzing graphs. We observe that graph trimming is essentially a kind of arc-consistency problem, and AC-3, AC-4, and AC-6 are the most relevant arc-consistency algorithms for application to graph trimming. The existing parallel graph trimming methods require worst-case \(\mathcal O(nm)\) time and worst-case \(\mathcal O(n)\) space for graphs with n vertices and m edges. We call these parallel AC-3-based as they are much like the AC-3 algorithm. In this work, we propose AC-4-based and AC-6-based trimming methods. That is, AC-4-based trimming has an improved worst-case time of \(\mathcal O(n+m)\) but requires worst-case space of \(\mathcal O(n+m)\); compared with AC-4-based trimming, AC-6-based has the same worst-case time of \(\mathcal O(n+m)\) but an improved worst-case space of \(\mathcal O(n)\). We parallelize the AC-4-based and AC-6-based algorithms to be suitable for shared-memory multi-core machines. The algorithms are designed to minimize synchronization overhead. For these algorithms, we also prove the correctness and analyze time complexities with the work-depth model. In experiments, we compare these three parallel trimming algorithms over a variety of real and synthetic graphs on a multi-core machine, where each core corresponds to a worker. Specifically, for the maximum number of traversed edges per worker by using 16 workers, AC-3-based traverses up to 58.3 and 36.5 times more edges than AC-6-based trimming and AC-4-based trimming, respectively. That is, AC-6-based trimming traverses much fewer edges than other methods, which is meaningful especially for implicit graphs. In particular, for the practical running time, AC-6-based trimming achieves high speedups over graphs with a large portion of trimmable vertices.

Message Passing Algorithms for Semiring-Based and Valued Constraint Satisfaction Problems

Local consistency algorithms, like arc consistency (AC) algorithms, are polynomial-time algorithms that prune the search space of constraint satisfaction problems (CSPs). In this paper, we present connections between message passing algorithms and AC for semiring-based CSPs (SCSPs) and valued CSPs (VCSPs), two well-established frameworks that generalize CSPs. Message passing algorithms are well known distributed search algorithms for solving many combinatorial problems in artificial intelligence, probabilistic reasoning, and information theory. However, the relationship between message passing algorithms and SCSPs or VCSPs still remains understudied. Towards this end, we propose the best-O message passing (BOMP) algorithm for SCSPs and VCSPs. We prove that, unlike other standard message passing algorithms which are in general not guaranteed to converge, the BOMP algorithm guarantees convergence for SCSPs and specific subclasses of VCSPs. We also theoretically study the relationship between the BOMP algorithm and AC on SCSPs, and empirically study the quality of the solutions produced by the BOMP algorithm for VCSPs.

Generalized Arc Consistency Algorithms for Table Constraints: A Summary of Algorithmic Ideas

Constraint Programming is a powerful paradigm to model and solve combinatorial problems. While there are many kinds of constraints, the table constraint (also called a CSP) is perhaps the most significant—being the most well-studied and has the ability to encode any other constraints defined on finite variables. Thus, designing efficient filtering algorithms on table constraints has attracted significant research efforts. In turn, there have been great improvements in efficiency over time with the evolution and development of AC and GAC algorithms. In this paper, we survey the existing filtering algorithms for table constraint focusing on historically important ideas and recent successful techniques shown to be effective.

Geometry-based distributed arc-consistency method for multiagent planning and scheduling

This research focuses on building a distributed algorithm for planning and scheduling multiple agents to help people deal with events beyond their cognitive capacity, such as car assembly, factory management, spacecraft constellation, etc. We address not only the efficiency of the algorithm but also communication and the individual privacy. As to reason over the problems with multiple agents which are distributed but interconnected, a formal account of the Action-centric Multiagent Simple Temporal Problem (AMSTP) is put forward using the representation of geometries. The key technique we build on is a novel distributed arc-consistency algorithm centered by the geometric method called GDAC, which pays attention to how an agent’s local subproblem affects other agents’ subproblems. The GDAC is based on geometries taking the action rather than the timepoint as a variable, which can deal with continuous intervals and decrease the number of variables. Comprehensive experiments are run and the proposed technique outperforms the competitor and shows considerable merit compared to the centralized algorithm.

A new distributed algorithm for efficient generalized arc-consistency propagation

Generalized arc-consistency propagation is predominantly used in constraint solvers to efficiently prune the search space when solving constraint satisfaction problems. Although many practical applications can be modelled as distributed constraint satisfaction problems, no distributed arc-consistency algorithms so far have considered the privacy of individual agents. In this paper, we propose a new distributed arc-consistency algorithm, called $$\mathsf {DisAC3.1}$$ , which leaks less private information of agents than existing distributed arc-consistency algorithms. In particular, $$\mathsf {DisAC3.1}$$ uses a novel termination determination mechanism, which allows the agents to share domains, constraints and communication addresses only with relevant agents. We further extend $$\mathsf {DisAC3.1}$$ to $$\mathsf {DisGAC3.1}$$ , which is the first distributed algorithm that enforces generalized arc-consistency on k-ary ( $$k\ge 2$$ ) constraint satisfaction problems. Theoretical analyses show that our algorithms are efficient in both time and space. Experiments also demonstrate that $$\mathsf {DisAC3.1}$$ outperforms the state-of-the-art distributed arc-consistency algorithm and that $$\mathsf {DisGAC3.1}$$ ’s performance scales linearly in the number of agents.

On the speed of constraint propagation and the time complexity of arc consistency testing

Establishing arc consistency on two relational structures is one of the most popular heuristics for the constraint satisfaction problem. We aim at determining the time complexity of arc consistency testing. The input structures G and H can be supposed to be connected colored graphs, as the general problem reduces to this particular case. We first observe the upper bound O(e(G)v(H)+v(G)e(H)), which implies the bound O(e(G)e(H)) in terms of the number of edges and the bound O((v(G)+v(H))3) in terms of the number of vertices. We then show that both bounds are tight as long as an arc consistency algorithm is based on constraint propagation (as all current algorithms are). Our lower bounds are based on examples of slow constraint propagation. We measure the speed of constraint propagation observed on a pair G,H by the size of a combinatorial proof that Spoiler wins the existential 2-pebble game on G,H.

Generalized support and formal development of constraint propagators

Constraint programming is a family of techniques for solving combinatorial problems, where the problem is modelled as a set of decision variables (typically with finite domains) and a set of constraints that express relations among the decision variables. One key concept in constraint programming is propagation: reasoning on a constraint or set of constraints to derive new facts, typically to remove values from the domains of decision variables. Specialized propagation algorithms (propagators) exist for many classes of constraints.The concept of support is pervasive in the design of propagators. Traditionally, when a domain value ceases to have support, it may be removed because it takes part in no solutions. Arc-consistency algorithms such as AC2001 make use of support in the form of a single domain value. GAC algorithms such as GAC-Schema use a tuple of values to support each literal. We generalize these notions of support in two ways. First, we allow a set of tuples to act as support. Second, the supported object is generalized from a set of literals (GAC-Schema) to an entire constraint or any part of it.We design a methodology for developing correct propagators using generalized support. A constraint is expressed as a family of support properties, which may be proven correct against the formal semantics of the constraint. We show how to derive correct propagators from the constructive proofs of the support properties. The framework is carefully designed to allow efficient algorithms to be produced. Derived algorithms may make use of dynamic literal triggers or watched literals for efficiency. Finally, three case studies of deriving efficient algorithms are given.

Saving constraint checks in maintaining coarse-grained generalized arc consistency

Constraint check plays a central role in establishing generalized arc consistency which is widely used to solve constraint satisfaction problems. In this paper, we propose a new generalized arc consistency algorithm, called GTR, which ensures that the tuples that have been checked to be allowed by a constraint will never be checked again. For each constraint, GTR maintains a dynamic list of the tuples that were checked to be allowed by this constraint and check their validities to identify some values with supports. It is equipped with a mechanism avoiding redundant validity checks. The basic GAC3 algorithm is employed to find a support for the rest values and to add new tuples to the dynamic list. The experiments show that maintaining GTR during search saves a number of constraint checks. It also brings some improvements over cpu time while solving some CSPs with tight constraints.

A Constraint Network Model and Parallel Arc Consistency Algorithms Based on GPU

A Constraint Network Model and Parallel Arc Consistency Algorithms Based on GPU

Narrowing Support Searching Range in Maintaining Arc Consistency for Solving Constraint Satisfaction Problems

Arc consistency is the most popular filtering technique for solving constraint satisfaction problems. Constraint check plays a central role in establishing arc consistency. In this paper, we propose a method to save constraint checks in maintaining coarse-grained arc consistency during backtracking search for solving the constraint satisfaction problems. We reduce the support searching range by utilizing the information generated by an AC3.1 algorithm at preprocessing step. Compared with the existing maintaining arc consistency (MAC) algorithms, the proposed MAC3 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">be</sub> algorithm saves constraint checks without maintaining additional data structures at each search tree node. Our experimental results show that MAC3 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">be</sub> saves both constraint checks and CPU time while solving some benchmark problems.

Revisiting restricted path consistency

Restricted path consistency (RPC) is a strong local consistency for binary constraints that was proposed 20 years ago and was identified as a promising alternative to arc consistency (AC) in an early experimental study of local consistencies for binary constraints. However, in contrast to other strong local consistencies such as singleton arc consistency (SAC) and max restricted path consistency (maxRPC), it has been neglected since then. In this paper we revisit RPC. First we propose RPC3, a new RPC algorithm that is very easy to implement and can be efficiently applied throughout search. Then we perform a wide experimental study of RPC3 and a light version that achieves an approximation of RPC, comparing them to state-of-the-art AC and maxRPC algorithms. Experimental results obtained under various solver settings, regarding the branching scheme, the variable ordering heuristic, and restarts, clearly show that light RPC is by far more efficient than both AC and maxRPC when applied throughout search. We also examine the experimental behaviour of an algorithm for a simple extension of RCP called 2-RPC, showing that it is competitive with RPC3, and better than both AC and maxRPC. Overall, our results strongly suggest that it is time to reconsider the established perception that MAC is the best general purpose method for solving binary CSPs.

An optimal arc consistency algorithm for a particular case of sequence constraint

The AtMostSeqCard constraint is the conjunction of a cardinality constraint on a sequence of n variables and of nÂ?Â?Â?qÂ?+Â?1 constraints AtMost u on each subsequence of size q. This constraint is useful in car-sequencing and crew-rostering problems. In van Hoeve et al. (Constraints 14(2):273---292, 2009), two algorithms designed for the AmongSeq constraint were adapted to this constraint with an O(2 q n) and O(n 3) worst case time complexity, respectively. In Maher et al. (2008), another algorithm similarly adaptable to filter the AtMostSeqCard constraint with a time complexity of O(n 2) was proposed. In this paper, we introduce an algorithm for achieving arc consistency on the AtMostSeqCard constraint with an O(n) (hence optimal) worst case time complexity. Next, we show that this algorithm can be easily modified to achieve arc consistency on some extensions of this constraint. In particular, the conjunction of a set of m AtMostSeqCard constraints sharing the same scope can be filtered in O(nm). We then empirically study the efficiency of our propagator on instances of the car-sequencing and crew-rostering problems.

New Algorithms for Bidirectional Singleton Arc Consistency

Bidirectional singleton arc consistency (BiSAC) which is an extended singleton arc consistency (SAC) has been proposed recently. The first contribution of this paper is to propose and prove two theorems of BiSAC theoretically (one is a property of BiSAC and the other is the property of allowing the deletion of some BiSAC-inconsistent values). Secondly, based on these properties we present two algorithms, denoted by BiSAC-DF and BiSAC-DP, to enforce BiSAC. Also, we prove their correctness and analyze the space and time complexity of them in detail. Besides, for special circumstances, we show that BiSAC-DF admits a worst-case time complexity inO(en2d4)and a best one inO(en2d3)when the problem is an already BiSAC, while BiSAC-DP also has the same best one when the tightness is small. Finally, experiments on a wide range of CSP instances show BiSAC-DF and BiSAC-DP are usually around one order of magnitude faster than the existing BiSAC-1. For some special instances, BiSAC-DP is about two orders of magnitude efficient.

Improving Coarse-Grained Arc Consistency Algorithms in Solving Constraint Satisfaction Problems

Constraint satisfaction problems have been widely investigated in artificial intelligence area.This paper investigates whether the coarse-grained maintaining arc is consistent,which is used to solve CSPs.The study has found that during the search for a solution,there are ineffective revisions,which revise the arcs that point to assigned variables.This study has proved that such revisions are redundant and proposed a method to avoid such redundant revisions.The paper gives the improved basic frame for the coarse-grained arc consistency algorithms,named AC3_frame_ARR.The new frame can be applied to improve all the coarse-grained AC algorithms.The experimental results show that the improved algorithms can save at most 80% revisions and 40% CPU time.

A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems

A fine-grained arc-consistency algorithm for non-normalized constraint satisfaction problems Constraint programming is a powerful software technology for solving numerous real-life problems. Many of these problems can be modeled as Constraint Satisfaction Problems (CSPs) and solved using constraint programming techniques. However, solving a CSP is NP-complete so filtering techniques to reduce the search space are still necessary. Arc-consistency algorithms are widely used to prune the search space. The concept of arc-consistency is bidirectional, i.e., it must be ensured in both directions of the constraint (direct and inverse constraints). Two of the most well-known and frequently used arc-consistency algorithms for filtering CSPs are AC3 and AC4. These algorithms repeatedly carry out revisions and require support checks for identifying and deleting all unsupported values from the domains. Nevertheless, many revisions are ineffective, i.e., they cannot delete any value and consume a lot of checks and time. In this paper, we present AC4-OP, an optimized version of AC4 that manages the binary and non-normalized constraints in only one direction, storing the inverse founded supports for their later evaluation. Thus, it reduces the propagation phase avoiding unnecessary or ineffective checking. The use of AC4-OP reduces the number of constraint checks by 50% while pruning the same search space as AC4. The evaluation section shows the improvement of AC4-OP over AC4, AC6 and AC7 in random and non-normalized instances.

Local arc consistency for non-invertible semirings, with an application to multi-objective optimization

We extend algorithms for local arc consistency proposed in the literature in order to deal with (absorptive) semirings that may not be invertible. As a consequence, these consistency algorithms can be used as a pre-processing procedure in soft Constraint Satisfaction Problems (CSPs) defined over a larger class of semirings, such as those obtained from the Cartesian product of two (or more) semirings. One important instance of this class of semirings is adopted for multi-objective CSPs. First, we show how a semiring can be transformed into a novel one where the + operator is instantiated with the least common divisor (LCD) between the elements of the original semiring. The LCD value corresponds to the amount we can “safely move” from the binary constraint to the unary one in the arc consistency algorithm. We then propose a local arc consistency algorithm which takes advantage of this LCD operator.

An MDD-based generalized arc consistency algorithm for positive and negative table constraints and some global constraints

A table constraint is explicitly represented as its set of solutions or non-solutions. This ad hoc (or extensional) representation may require space exponential to the arity of the constraint, making enforcing GAC expensive. In this paper, we address the space and time inefficiencies simultaneously by presenting the mddc constraint. mddc is a global constraint that represents its (non-)solutions with a multi-valued decision diagram (MDD). The MDD-based representation has the advantage that it can be exponentially smaller than a table. The associated GAC algorithm (called mddc) has time complexity linear to the size of the MDD, and achieves full incrementality in constant time. In addition, we show how to convert a positive or negative table constraint into an mddc constraint in time linear to the size of the table. Our experiments on structured problems, car sequencing and still-life, show that mddc is also a fast GAC algorithm for some global constraints such as sequence and regular. We also show that mddc is faster than the state-of-the-art generic GAC algorithms in Gent et al. (2007), Lecoutre and Szymanek (2006), Lhomme and Regin (2005) for table constraint.

Efficient algorithms for singleton arc consistency

In this paper, we propose two original and efficient approaches for enforcing singleton arc consistency. In the first one, the data structures used to enforce arc consistency are shared between all subproblems where a domain is reduced to a singleton. This new algorithm is not optimal but it requires far less space and is often more efficient in practice than the optimal algorithm SAC-Opt. In the second approach, we perform several runs of a greedy search (where at each step, arc consistency is maintained), possibly detecting the singleton arc consistency of several values in one run. It is an original illustration of applying inference (i.e., establishing singleton arc consistency) by search. Using a greedy search allows benefiting from the incrementality of arc consistency, learning relevant information from conflicts and, potentially finding solution(s) during the inference process. We present extensive experiments that show the benefit of our two approaches.