# The complexity analysis of solving the max-product fuzzy relation equation with LU decomposition

## Abstract

\(L\circ U\)-factorization was recently used to solve the max-product fuzzy relation equation by Molai (Inf Sci 234:86–96, 2013). Considering the forward and backward substitutions play an important role in this method, this paper firstly amend the forward and backward substitutions for solving max-product fuzzy relation equation with \(L\circ U\)-factorization. And then, the computational complexities of improved forward and backward substitutions are analyzed in detail. Finally, we find that the \(L\circ U\)-factorization acts as splitting an irredundant covering of max-product fuzzy relation equation into two parts. It therefore cannot change the fact that finding the solutions of max-product fuzzy relation equation with \(L\circ U\)-factorization is an NP-hard problem. On the contrary, the computational expense will linearly increase with the number of minimal solutions of \(L\circ \mathbf {y}=\mathbf {b}\).

## Keywords

Fuzzy relation equation Max-product composition \(L\circ U\)-factorization Solutions set Computational complexity## Notes

### Acknowledgements

The authors are very appreciative to the anonymous referees and the Editor-in-Chief for their valuable comments that helped us to improve our paper quality. This work was supported by the National Natural Science Foundation of China (Grant No. 61673352).

### Compliance with ethical standards

### Conflict of interest

Author declares that he has no conflict of interest.

### Human and animal rights

This article does not contain any studies with human participants or animals performed by the authors.

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