Generalized Majority-Minority Operations are Tractable

“…While Mal'tsev and near unanimity operations are of quite different character, Dalmau in [22] managed to find a common generalization of them via the generalized majority-minority operation (see Example 4 for the definition). In a modification of the algorithm presented in [13], Dalmau shows in [22] that any finite algebra that has a GMM term is tractable.…”

Section: Definition 5 ([2])mentioning

confidence: 99%

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Recent Results on the Algebraic Approach to the CSP

Bulatov

Valeriote

2008

Complexity of Constraints

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“…A unified approach to Mal'tsev operations and near-unanimity operations, which generalises Proposition 41 and Proposition 46 is given in [29].…”

Section: Proposition 41 ([50]) For Any Constraint Language γ Over a mentioning

confidence: 99%

The Complexity of Constraint Languages

Cohen¹,

Jeavons²

2006

Foundations of Artificial Intelligence

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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

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Generalized Majority-Minority Operations are Tractable

Cited by 40 publications

References 27 publications

A rendezvous of logic, complexity, and algebra

A rendezvous of logic, complexity, and algebra

Recent Results on the Algebraic Approach to the CSP

The Complexity of Constraint Languages

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