The World of Combinatorial Fuzzy Problems and the Efficiency of Fuzzy Approximation Algorithms

TL;DR

This paper explores the complexity and efficiency of fuzzy algorithms for combinatorial problems, introducing models, complexity classes, and reducibility notions to analyze fuzzy problem-solving methods.

Contribution

It develops polynomial-time fuzzy algorithms, models fuzzy complexity classes, and introduces reducibility concepts for fuzzy problems, advancing understanding of fuzzy computational complexity.

Findings

Defined fuzzy complexity classes such as Fuzzy-FPA, Fuzzy-PA, Fuzzy-NPA, and Fuzzy-NPAO.

Designed polynomial-time fuzzy algorithms and fuzzy circuit models.

Established reducibility notions to compare fuzzy problem complexities.

Abstract

We re-examine a practical aspect of combinatorial fuzzy problems of various types, including search, counting, optimization, and decision problems. We are focused only on those fuzzy problems that take series of fuzzy input objects and produce fuzzy values. To solve such problems efficiently, we design fast fuzzy algorithms, which are modeled by polynomial-time deterministic fuzzy Turing machines equipped with read-only auxiliary tapes and write-only output tapes and also modeled by polynomial-size fuzzy circuits composed of fuzzy gates. We also introduce fuzzy proof verification systems to model the fuzzification of nondeterminism. Those models help us identify four complexity classes: Fuzzy-FPA of fuzzy functions, Fuzzy-PA and Fuzzy-NPA of fuzzy decision problems, and Fuzzy-NPAO of fuzzy optimization problems. Based on a relative approximation scheme targeting fuzzy membership degree, we formulate two notions of "reducibility" in order to compare the computational complexity of two fuzzy problems. These reducibility notions make it possible to locate the most difficult fuzzy problems in Fuzzy-NPA and in Fuzzy-NPAO.