Finite Domain Constraint Satisfaction Research Articles

The Complexity of Network Satisfaction Problems for Symmetric Relation Algebras with a Flexible Atom

Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is symmetric and has a fexible atom; in this case, the problem is NP-complete or in P. The classification task can be reduced to the case where A is integral. If a finite integral relation algebra has a flexible atom, then it has a normal representation B. We can then study the computational complexity of the network satisfaction problem of A using the universal-algebraic approach, via an analysis of the polymorphisms of B. We also use a Ramsey-type result of Nešetřil and Rödl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.

On the Descriptive Complexity of Temporal Constraint Satisfaction Problems

Finite-domain constraint satisfaction problems are either solvable by Datalog or not even expressible in fixed-point logic with counting. The border between the two regimes can be described by a universal-algebraic minor condition. For infinite-domain constraint satisfaction problems (CSPs), the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (ℚ;<); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the mod-2 rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.

Network Satisfaction for Symmetric Relation Algebras with a Flexible Atom

Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation B. We can then study the computational complexity of the network satisfaction problem of A using the universal-algebraic approach, via an analysis of the polymorphisms of B. We also use a Ramsey-type result of Nešetřil and Rödl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.

Topology Is Irrelevant (In a Dichotomy Conjecture for Infinite Domain Constraint Satisfaction Problems)

The tractability conjecture for finite domain constraint satisfaction problems (CSPs) stated that such CSPs are solvable in polynomial time whenever there is no natural reduction, in some precise t...

Distance constraint satisfaction problems

We study the complexity of constraint satisfaction problems for templates Γ over the integers where the relations are first-order definable from the successor function. In the case that Γ is locally finite (i.e., the Gaifman graph of Γ has finite degree), we show that Γ is homomorphically equivalent to a structure with one of two classes of polymorphisms (which we call modular max and modular min) and the CSP for Γ can be solved in polynomial time, or Γ is homomorphically equivalent to a finite transitive structure, or the CSP for Γ is NP-complete. Assuming a widely believed conjecture from finite domain constraint satisfaction (we require the tractability conjecture by Bulatov, Jeavons and Krokhin in the special case of transitive finite templates), this proves that those CSPs have a complexity dichotomy, that is, are either in P or NP-complete.

Enhanced Spreadsheet Computing with Finite Domain Constraint Satisfaction

The spreadsheet application is among the most widely used computing tools in modern society. It provides excellent usability and usefulness, and it easily enables a non-programmer to perform programming-like tasks in a visual tabular "pen and paper" approach. However, spreadsheets are mostly limited to bookkeeping-like applications due to their mono-directional data flow. This paper shows how the spreadsheet computing paradigm is extended to break this limitation for solving constraint satisfaction problems. We present an enhanced spreadsheet system where finite-domain constraint solving is well supported in a visual environment. Furthermore, a spreadsheet-specific constraint language is constructed for general users to specify constraints among data cells in a declarative and scalable way. The new spreadsheet system significantly simplifies the development of many constraint-based applications using a visual tabular interface. Examples are given to illustrate the usability and usefulness of the extended spreadsheet paradigm. KEYWORDS: Spreadsheet computing, Finite-domain constraint satisfaction, Constraint logic programming

Enhanced Spreadsheet Computing with Finite Domain Constraint Satisfaction

Enhanced Spreadsheet Computing with Finite Domain Constraint Satisfaction

Constraint satisfaction tractability from semi-lattice operations on infinite sets

A famous result by Jeavons, Cohen, and Gyssens shows that every Constraint Satisfaction Problem (CSP) where the constraints are preserved by a semi-lattice operation can be solved in polynomial time. This is one of the basic facts for the so-called universal algebraic approach to a systematic theory of tractability and hardness in finite domain constraint satisfaction. Not surprisingly, the theorem of Jeavons et al. fails for arbitrary infinite domain CSPs. Many CSPs of practical interest, though, and in particular those CSPs that are motivated by qualitative reasoning calculi from artificial intelligence, can be formulated with constraint languages that are rather well-behaved from a model-theoretic point of view. In particular, the automorphism group of these constraint languages tends to be large in the sense that the number of orbits of n -subsets of the automorphism group is bounded by some function in n . In this article we present a generalization of the theorem by Jeavons et al. to infinite domain CSPs where the number of orbits of n -subsets grows subexponentially in n , and prove that preservation under a semi-lattice operation for such CSPs implies polynomial-time tractability. Unlike the result of Jeavons et al., this includes CSPs that cannot be solved by Datalog.

Towards Interesting Patterns of Hard CSPs with Functional Constraints

The hardness of finite domain Constraint Satisfaction Problems (CSPs) is an important research topic in Constraint Programming (CP) community. In this paper, we study the association rule mining techniques together with rule deduction and propose a cascaded approach to extract interesting patterns of hard CSPs with functional constraints. Specifically, we generate random CSPs, collect controlling parameters and hardness characteristics by solving all the CSP instances. Afterwards, we apply association rule mining with rule deduction on the collected data set and further extract interesting patterns of the hardness of the randomly generated CSPs. As far as we know, this problem is investigated with data mining techniques for the first time.

Image interpretation with a conceptual graph: Labeling over-segmented images and detection of unexpected objects

The labeling of the regions of a segmented image according to a semantic representation (ontology) is usually associated with the notion of understanding. The high combinatorial aspect of this problem can be reduced with local checking of constraints between the elements of the ontology. In the classical definition of Finite Domain Constraint Satisfaction Problem, it is assumed that the matching problem between regions and labels is bijective. Unfortunately, in image interpretation the matching problem is often non-univocal. Indeed, images are often over-segmented: one object is made up of several regions. This non-univocal matching between data and a conceptual graph was not possible until a decisive step was accomplished by the introduction of arc consistency with bilevel constraint ( FDCSP BC ). However, this extension is only adequate for a matching corresponding to surjective functions. In medical image analysis, the case of non-functional relations is often encountered, for example, when an unexpected object like a tumor appears. In this case, the data cannot be mapped to the conceptual graph, with a classical approach. In this paper we propose an extension of the FDCSP BC to solve the constraint satisfaction problem for non-functional relations.

Backjump-based backtracking for constraint satisfaction problems

The performance of backtracking algorithms for solving finite-domain constraint satisfaction problems can be improved substantially by look-back and look-ahead methods. Look-back techniques extract information by analyzing failing search paths that are terminated by dead-ends. Look-ahead techniques use constraint propagation algorithms to avoid such dead-ends altogether. This paper describes a number of look-back variants including backjumping and constraint recording which recognize and avoid some unnecessary explorations of the search space. The last portion of the paper gives an overview of look-ahead methods such as forward checking and dynamic variable ordering, and discusses their combination with backjumping.

Learning solution preferences in constraint problems

. Usually, not all the solutions of a finite domain constraint satisfaction problem (CSP) are equally desirable: some of them may be preferred to others. However, classical CSPs do not allow for this more informative kind of knowledge representation. On the other hand, semiring-based CSPs (SCSPs), where a value is associated with each tuple in each constraint, generate solutions with a corresponding value attached that can be interpreted as the level of preference of that solution. Sometimes, however, even standard SCSPs are not enough, since one may know preferences over some of the solutions but have no idea on how to code this knowledge into the SCSP. In this paper we consider this situationand propose to address it by first defining a classical CSP and giving some examples of solution preferences, and then learning the corresponding SCSP that behaves as the initial CSP (that is, it has the same solutions) and matches the preferences specified in the examples. In other words, we use the examples as the training set, and we employ a learning scheme to adjust the values to be attached to the constraint tuples, such that the resulting solution preferences coincide with the examples. In this way, we make the SCSP framework more flexible, since it can be used also when it is difficult to assign values to tuples and instead it is easier to rate some of the solutions.

Constraint satisfaction problem with bilevel constraint: application to interpretation of over-segmented images

In classical finite-domain constraint satisfaction problems, the assumption made is that only one value is associated with only one variable. For example, in pattern recognition one variable is associated with only one segmented region. However, in practice, regions are often oversegmented which results in failure of any one to one mapping. This paper proposes a definition of finite-domain constraint satisfaction problems with bilevel constraints in order to take into account a many to one relation between the values and the variables. The additional level of constraint concerns the data assigned to the same complex variable. Then, we give a definition of the arc-consistency problem for bilevel constraint satisfaction checking. A new algorithm for arc consistency to deal with these problems is presented as well. This extension of the arc-consistency algorithm retains its good properties and has a time complexity in O( en 3 d 2) in the worst case. This algorithm was tested on medical images. These tests demonstrate its reliability in correctly identifying the segmented regions even when the image is over-segmented.