Algorithms For Constraint Satisfaction Problems Research Articles

Many hard examples in exact phase transitions

This paper analyzes the resolution complexity of two random constraint satisfaction problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.

When Ants Attack: Ant Algorithms for Constraint Satisfaction Problems

We describe an ant algorithm for solving constraint problems (Solnon 2002, IEEE Transactions on Evolutionary Computation 6(4): 347---357). We devise a number of variants and carry out experiments. Our preliminary results suggest that the best way to deposit pheromone and the best heuristics for state transitions may differ from current practice

Lossy Data Compression with Random Gates

We introduce a new protocol for a lossy data compression algorithm which is based on constraint satisfaction gates. We show that the theoretical capacity of algorithms built from standard parity-check gates converges exponentially fast to the Shannon's bound when the number of variables seen by each gate increases. We then generalize this approach by introducing random gates. They have theoretical performances nearly as good as parity checks, but they offer the great advantage that the encoding can be done in linear time using the survey inspired decimation algorithm, a powerful algorithm for constraint satisfaction problems derived from statistical physics.

A polynomial algorithm for continuous non-binary disjunctive CSPs: extended DLRs

Nowadays, many real problems can be modelled as Constraint Satisfaction Problems (CSPs). Some CSPs are considered non-binary disjunctive CSPs. Many researchers study the problems of deciding consistency for Disjunctive Linear Relations (DLRs). In this paper, we propose a new class of constraints called Extended DLRs consisting of disjunctions of linear inequalities, linear disequations and non-linear disequations. This new class of constraints extends the class of DLRs. We propose a heuristic algorithm called DPOLYSA that solves Extended DLRs, as a non-binary disjunctive CSP solver. This proposal works on a polyhedron whose vertices are also polyhedra that represent the nondisjunctive problems. We also present a statistical preprocessing step which translates the disjunctive problem into a non-disjunctive and ordered one in each step.

Reduction operations in fuzzy or valued constraint satisfaction

In the constraint satisfaction problem (CSP) knowledge about a relation on n variables is given in the form of constraints on subsets of the variables. A solution to the CSP is simply an instantiation of the n variables which satisfies all the constraints. The fuzzy constraint satisfaction problem (FCSP) is a generalisation of the CSP to the case in which the constraints are not categorical but represent preferences in the form of fuzzy sets. The FCSP is then an optimisation problem in the space of all possible instantiations of the n variables. It has potentially many more applications in real-world problems than the CSP. When the aim is to maximise the value of the most violated constraint, then the FCSP can be solved by solving a logarithmic number of CSPs. Fuzzy k-consistency can even be established by an algorithm whose worst-case complexity is identical to that of an optimal k-consistency algorithm for CSPs. When the operator used to aggregate constraint values is strictly monotonic (as is the case in MAX-CSP, for example) the classic operation of arc consistency has to be completely redefined. We show that an ordered version of arc consistency exists for all FCSPs with a strictly monotonic aggregation operator, and that, in the case of MAX-CSP, it can be established by an algorithm whose worst-case complexity is identical to that of the optimal arc consistency algorithm for CSPs. Neighbourhood substitution in CSPs can be generalised to fuzzy neighbourhood substitution in FCSPs, whether the aggregation operator is strictly monotonic or idempotent. We give an algorithm for applying fuzzy neighbourhood substitutions until convergence based on the corresponding algorithm for CSPs.

The Path repair algorithm

Abstract In this paper, we introduce a new solving algorithm for Constraint Satisfaction Problems the path-repair algorithm . The two main points of that algorithm are: it makes use of a repair algorithm (local search) as a basis and it works on a partial instantiation in order to be able to use filtering techniques. Different versions are presented and first experiments with both systematic and non systematic versions show promising results.

Using an Incomplete Version of Dynamic Backtracking for Graph Colouring

Abstract This paper investigates a simple, constructive search strategy: an incomplete version of Ginsberg's Dynamic Backtracking which records no information during search, and randomly selects variables for backtracking. Combining this with forward checking and dynamic variable ordering heuristics we obtain algorithms for constraint satisfaction problems, and by adding branch-and-bound we obtain optimization algorithms. Testing these on several classes of 1000-vertex graph colouring problems we find colourings which are competitive with those of specialized algorithms: a 92-colouring for a random graph (p = 0.5), hidden k-colourings in equipartite graphs (p = 0.5) for k ≤ 60 and improved results for geometric graphs. An interesting aspect is that most of our results were obtained using the precise opposite of the well-known first-fail principle.

Probe Order Backtracking

The algorithm for constraint-satisfaction problems, Probe Order Backtracking, has an average running time much faster than any previously analyzed algorithm under conditions where solutions are common. The algorithm uses a probing assignment (a preselected test assignment to unset variables) to help guide the search for a solution. If the problem is not satisfied when the unset variables are temporarily set to the probing assignment, the algorithm selects one of the relations which is not satisfied by the probing assignment and selects an unset variable which affects the value of that relation. It then does a backtracking (splitting) step, where it generates subproblems by setting the selected variable each possible way. Each subproblem is simplified and then solved recursively. For random problems with v variables, t clauses, and probability p that a literal appears in a clause, the average time for Probe Order Backtracking is no more than $v^n$ when $p\geq(\ln t)/v$ plus lower-order terms. The best previous result was $p\geq\sqrt{(\ln t)/v}$. When the algorithm is combined with an algorithm of Franco that makes selective use of resolution, the average time for solving random problems is no more than $v^n$ for all values of p when $t\leq O(n^{1/3}(v/\ln v)^{2/3})$. The best previous result was $t\leq O(n^{1/3}(v/\ln v)^{1/6})$. Probe Order Backtracking also runs in polynomial average time when $p\leq 1/v$, compared with the best previous result of $p\leq 1/(2v)$. With Probe Order Backtracking, the range of p that leads to more than polynomial time is much smaller than that for previously analyzed algorithms.

Scheduling as a multi-dimensional placement problem

This paper describes research in developing an algorithm for a constraint-satisfaction problem (CSP) which solves certain classes of scheduling problems by formulating them as multi-dimensional placement problems. The test domain is check-in counter scheduling at an international airport. In this airport the counters are managed and allocated centrally by the local aviation authority. This scheduling problem is complicated by the fact that the resource requirement is stochastic—passenger arrival is a non-stationary probability distribution; the resources are not uniform—the physical location and available free space for queuing are different for different counters; and the resources to be allocated have other associated resources with their own sets of constraints—counters have associated baggage belts that have different capacity constraints. The approach presented combines simulation techniques with constraint-based reasoning to handle the stochastic nature of the problem. Scheduling requirements are modelled as multi-dimensional objects. The CSP algorithm presented here treats scheduling as a process of kneading and moulding these multi-dimensional objects into unallocated free space until a place is found for each of these objects. The constraints of the problem define how much the shape can be moulded.

UNCERTAINTY REASONING IN ASTRONOMY--APPLICATIONS TO CLASSIFICATION AND OPTIMAL TELESCOPE SCHEDULING

ABSTRACT This thesis addresses two topics: the "selection of interesting objects from astronomical catalogues" and the "design of optimised observation schedules." Part I of the thesis deals with the question of how to optimally select interesting objects from astronomical databases. To this end we use a trainable statistical pattern recognition system. The chapter "Supervised and unsupervised classification --- The case of IRAS point sources" provides an short introduction to and review of supervised and un\\-supervised classification. The chapter "Mapping the extragalactic sky with IRAS PSC data" describes the selection of galaxy candidates from the IRAS Point Source Catalog. The distributions of the infrared colours are approximated as multivariate Gaussian distributions. The selection is carried out by statistical classifiers based on the maximum likelihood (ML) and maximum a posteriori (MAP) principles. The chapter "Young stellar objects in the IRAS Point Source Catalog" uses essentially the same methods for selecting young stellar object (YSO) candidates from the IRAS Point Source Catalog. In this case the information provided by the infrared colours alone is insufficient for a good discrimination of YSO candidates against other objects. However with the help of additional uncertain information provided by the spatial clustering properties of the objects on the sky, the selection can be significantly improved. The chapter "Statistical inferencing in the design of a sky-survey project" attempts to bridge the wide gap remaining between a selection of interesting candidates from some catalogue and the design of a suitably optimised strategy for follow-up observations. Part II of this thesis deals with the problem of how to create and optimise a long-term schedule for an astronomical telescope. Specifically the case of scheduling the Hubble Space Telescope (HST) observations is considered. It is shown how an artificial neural network with a novel stochastic dynamics can be used to efficiently solve the underlying constraint satisfaction problem (CSP). The chapter "The processing of HST observing programmes" introduces the HST long range planning problem and explains why the scheduling of pointed observations with this astronomical satellite in a low Earth orbit is difficult. The chapter "Connectionism and neural networks" provides some conceptual background on artificial neural networks. It is shown how a simple scheduling problem can be mapped onto the topology of an artificial neural network. Adding a suitable network dynamics turns such a network into an efficient problem solver. The chapter "A discrete stochastic neural network algorithm for constraint satisfaction problems" describes the novel "guarded discrete stochastic" (GDS) network algorithm, a relative of the standard Hopfield-Tank network, which is surprisingly efficient in finding solutions to large constraint satisfaction problems even on serial machines. The chapter "Scheduling with neural networks --- The case of Hubble Space Telescope" finally describes how the GDS network is integrated into Spike, the scheduling system developed at the Space Telescope Science Institute, Baltimore, for creating long range HST observing schedules. The methodology described here is quite general and is applicable to scheduling other ground-based or space-born astronomical observatories.

Experimental evaluation of preprocessing algorithms for constraint satisfaction problems

This paper presents an experimental evaluation of two orthogonal schemes for pre-processing constraint satisfaction problems (CSPs). The first of these schemes involves a class of local consistency techniques that includes directional arc consistency, directional path consistency, and adaptive consistency. The other scheme concerns the prearrangement of variables in a linear order to facilitate an efficient search. In the first series of experiments, we evaluated the effect of each of the local consistency techniques on backtracking and backjumping. Surprisingly, although adaptive consistency has the best worst-case complexity bounds, we have found that it exhibits the worst performance, unless the constraint graph was very sparse. Directional arc consistency (followed by either backjumping or backtracking) and backjumping (without any preprocessing) outperformed all other techniques: moreover, the former dominated the latter in computationally intensive situations. The second series of experiments suggests that maximum cardinality and minimum width are the best preordering (i.e., static ordering) strategies, while dynamic search rearrangement is superior to all the preorderings studied.

The complexity of constraint satisfaction revisited

This paper is a retrospective account of some of the developments leading up to, and ensuing from, the analysis of the complexity of some polynomial network consistency algorithms for constraint satisfaction problems.

Algorithms for constraint-satisfaction problems

Algorithms for constraint-satisfaction problems

Algorithms for constraint-satisfaction problems: a survey

A large number of problems in AI and other areas of computer science can be viewed as special cases of the constraint-satisfaction problem. Some examples are machine vision, belief maintenance, scheduling, temporal reasoning, graph problems, floor plan design, the planning of genetic experiments, and the satisfiability problem. A number of different approaches have been developed for solving these problems. Some of them use constraint propagation to simplify the original problem. Others use backtracking to directly search for possible solutions. Some are a combination of these two techniques. This article overviews many of these approaches in a tutorial fashion.

The complexity of some polynomial network consistency algorithms for constraint satisfaction problems

Constraint satisfaction problems play a central role in artificial intelligence. A class of network consistency algorithms for eliminating local inconsistencies in such problems has previously been described. We analyze the time complexity of several node, arc and path consistency algorithms and prove that arc consistency is achievable in time linear in the number of binary constraints. The Waltz filtering algorithm is a special case of the arc consistency algorithm. In the edge labelling computational vision application the constraint graph is planar and so the time complexity is linear in the number of variables.