M-best solutions for a class of fuzzy constraint satisfaction problems

TL;DR

This paper introduces a polynomial-time algorithm for finding the top d solutions in a generalized fuzzy constraint satisfaction problem, leveraging invariants and polymorphisms in (min,max) semirings without requiring explicit knowledge of the majority operator.

Contribution

It extends the classic CSP framework to fuzzy relations using multivalued membership functions and non-uniform polymorphisms, providing a novel polynomial-time solution method.

Findings

Algorithm finds d most admissible solutions efficiently

Works without knowing the majority operator in advance

Discards problems without a majority polymorphism

Abstract

The article considers one of the possible generalizations of constraint satisfaction problems where relations are replaced by multivalued membership functions. In this case operations of disjunction and conjunction are replaced by maximum and minimum, and consistency of a solution becomes multivalued rather than binary. The article studies the problem of finding d most admissible solutions for a given d. A tractable subclass of these problems is defined by the concepts of invariants and polymorphisms similar to the classic constraint satisfaction approach. These concepts are adapted in two ways. Firstly, the correspondence of "invariant-polymorphism" is generalized to (min,max) semirings. Secondly, we consider non-uniform polymorphisms, where each variable has its own operator, in contrast to the case of one operator common for all variables. The article describes an algorithm that finds $d$ most admissible solutions in polynomial time, provided that the problem is invariant with respect to some non-uniform majority operator. It is essential that this operator needs not to be known for the algorithm to work. Moreover, even a guarantee for the existence of such an operator is not necessary. The algorithm either finds the solution or discards the problem. The latter is possible only if the problem has no majority polymorphism.