Tractability in constraint satisfaction problems: a survey

Abstract

Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. What is tractability?The idea that an algorithm is efficient if its time complexity is a polynomial function of the size of its input can be traced back to pioneering work of Cobham [45] and Edmonds [83], but the foundations of complexity theory are based on the seminal work of Cook [52] and Karp [118]. A computational decision problem (such as the constraint satisfaction problem or CSP) consists of a generic instance (in the case of the CSP, a set of variables, their

The Power of Arc Consistency for CSPs Defined by Partially-Ordered Forbidden Patterns

“…The complexity of CSPs that are neither language-based nor structure-based, and thus are often called hybrid CSPs, is much less understood; see [9,20] for recent surveys. One approach to hybrid CSPs that has been rather successful studies the classes of CSPs defined by forbidden patterns; that is, by forbidding certain generic subinstances.…”

“…For instance, the well-known 2-SAT problem is a class of language-based CSPs on the Boolean domain {0, 1} with all constraint relations being binary, that is, of arity at most two.On the other side of the spectrum are structure-based CSPs, that is, classes of CSPs defined by the allowed interactions of the constraint scopes but with arbitrary constraint relations. Here the methods that have been successfully used to establish complete complexity classifications come from graph theory [28,33].The complexity of CSPs that are neither language-based nor structure-based, and thus are often called hybrid CSPs, is much less understood; see [9,20] for recent surveys. One approach to hybrid CSPs that has been rather successful studies the classes of CSPs defined by forbidden patterns; that is, by forbidding certain generic subinstances.…”

Extending Broken Triangles and Enhanced Value-Merging

“…However, by imposing some restrictions on the constraint scopes and/or relations, we can define tractable classes of instances which can be solved in polynomial time. The BTP (Broken Triangle Property) tractable class, is an important tractable class since it generalises certain previously known classes based exclusively on properties of the constraint scopes or the constraint relations and has been the inspiration for a new branch of research on tractable classes of CSPs based on forbidden patterns [1,4,9,11,18]. The Broken Triangle Property imposes the absence of so-called broken triangles.…”

Exploring Directional Path-Consistency for Solving Constraint Networks

“…Dechter and Pearl [15] showed that DPC is sufficient for enabling backtrack-free search for networks with constraint graphs of regular width 2. We consider the aforementioned question in the context of constraint languages, which is a widely adopted approach in the study of tractability of constraint satisfaction problems [8]. Specifically, we are interested in finding all binary constraint languages Γ such that the consistency of any constraint network defined over Γ can be decided by DPC.…”