The Complexity of Constraint Languages

Abstract

One of the most fundamental challenges in constraint programming is to understand the computational complexity of problems involving constraints. It has been shown that the class of all constraint satisfaction problem instances is NP-hard [71], so it is unlikely that efficient general-purpose algorithms exist for solving all forms of constraint problem. However, in many practical applications the instances that arise have special forms that enable them to be solved more efficiently [11,25,69,82].One way in which this occurs is that there is some special structure in the way that the constraints overlap and intersect each other. The natural theory for discussing the structure of such interaction between constraints is the mathematical theory of hypergraphs. Much work has been done in this area, and many tractable classes of constraint problems have been identified based on structural properties (see Chapter 5). There are strong parallels between this work and similar investigations into the structure of so-called conjunctive queries in relational databases [41,58].Another way in which constraint problems can be defined which are easier to solve than in the general case is when the types of constraints are limited. The natural theory for discussing the properties of constraint types is the mathematical theory of relations and their associated algebras. Again considerable progress has been made in this investigation over the past few years. For example, a complete characterisation of tractable constraint types is now known for both 2-element domains [85] and 3-element domains [14]. In addition, a number of novel efficient algorithms have been developed for solving particular types of constraint problems over both finite and infinite domains [3,8,16,25,26,28,63].In this chapter we will focus on the second approach. That is, we will investigate how the complexity of solving constraint problems varies with the types of constraints which are allowed. One fundamental open research problem in this area is to characterise exactly which types of constraints give rise to constraint problems which can be solved

Dualities for Constraint Satisfaction Problems

“…The algebraic approach to constraint satisfaction (see, e.g., [10][11][12]44]) has proved to be extremely successful. It provides a convenient dual language to analyse CSPs, and, more importantly, allows one to use powerful machinery from universal algebra.…”

“…Schaefer's celebrated dichotomy theorem for Boolean CSP can be restated (see, e.g., [10,12,44]) as follows. For a Boolean core structure B, if B has a semilattice polymorphism, or a majority polymorphism, or the affine polymorphism, then CSP(B) is in PTIME.…”

“…This is one of the two most general properties of relational structures B that are known to guarantee that the problem CSP(B) solvable in polynomial time, and the vast majority of such structures have bounded treewidth duality. The notion of bounded treewidth duality has strong links with methods of solving constraint satisfaction problems based on establishing local consistency (see, e.g., [12,18,37,28]). …”

“…The obtained problem is denoted by CSP(B). Examples of such problems include various versions of k-Sat, Graph Colouring, and Systems of Equations (see [12,28,36,44]). Strong motivation for studying this framework was given in [22] where it was shown that such problems can be used in attempts to identify a largest subclass of NP that avoids problems of intermediate complexity.…”

“…The most advanced approaches use logic (e.g., [42]), combinatorics (e.g., [26,28,46]), universal algebra (e.g., [6,9,10,12,35,44]), or combinations of those (e.g., [2,7,16,22,48]). In this survey, we will discuss a combinatorial idea that has a bearing on all the above problems, and has strong links with the three approaches -the idea of homomorphism duality.…”