Computing the Fuzzy Partition Corresponding to the Greatest Fuzzy Auto-Bisimulation of a Fuzzy Graph-Based Structure

Linh Anh Nguyen

Home/Archive/arXiv

arXiv · 2107.08542

Computing the Fuzzy Partition Corresponding to the Greatest Fuzzy Auto-Bisimulation of a Fuzzy Graph-Based Structure

Linh Anh Nguyen

View original source ↗Read source document ↗Export metadata ↓

Abstract

Fuzzy graph-based structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models, fuzzy social networks and fuzzy interpretations in fuzzy description logics are useful in various applications. Given two states, two actors or two individuals $x$ and $x'$ in such structures $G$ and $G'$, respectively, the similarity degree between them can be defined to be $Z(x,x')$, where $Z$ is the greatest fuzzy bisimulation between $G$ and $G'$ w.r.t. some t-norm-based fuzzy logic. Such a similarity measure has the Hennessy-Milner property of fuzzy bisimulations as a strong logical foundation. A fuzzy bisimulation between a fuzzy structure $G$ and itself is called a fuzzy auto-bisimulation of $G$. The greatest fuzzy auto-bisimulation of an image-finite fuzzy graph-based structure is a fuzzy equivalence relation. It is useful for classification and clustering. In this paper, we design an efficient algorithm with the complexity $O((m\log{l} + n)\log{n})$ for computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a finite fuzzy labeled graph $G$ under the G\"odel semantics, where $n$, $m$ and $l$ are the number of vertices, the number of non-zero edges and the number of different fuzzy degrees of edges of $G$, respectively. Our notion of fuzzy partition is novel, defined only for finite sets with respect to the G\"odel t-norm, with the aim to facilitate the computation of the greatest fuzzy auto-bisimulation. By using that algorithm, we also provide an algorithm with the complexity $O(m\cdot\log{l}\cdot\log{n} + n^2)$ for computing the greatest fuzzy bisimulation between two finite fuzzy labeled graphs under the G\"odel semantics. This latter algorithm is better (has a lower complexity order) than the previously known algorithms for the considered problem. Our algorithms can be restated for the other mentioned fuzzy graph-based structures.

Explore related subjects

cs.DS

Keep this discovery

BibTeX RIS

Linh Anh Nguyen. 2021-07-18. Computing the Fuzzy Partition Corresponding to the Greatest Fuzzy Auto-Bisimulation of a Fuzzy Graph-Based Structure. https://arxiv.org/abs/2107.08542

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗