Tolerant Bipartiteness Testing in Dense Graphs

• Main Authors: Ghosh, Arijit, Mishra, Gopinath, Raychaudhury, Rahul, Sen, Sayantan
• Format: Journal Article
• Language: English
• Published: 26-04-2022
• Subjects: Computer Science - Data Structures and Algorithms
• DOI: 10.48550/arxiv.2204.12397

Bipartite testing has been a central problem in the area of property testing
since its inception in the seminal work of Goldreich, Goldwasser and Ron
[FOCS'96 and JACM'98]. Though the non-tolerant version of bipartite testing has
been extensively studied in the literature, the tolerant variant is not well
understood. In this paper, we consider the following version of tolerant
bipartite testing: Given a parameter $\varepsilon \in (0,1)$ and access to the
adjacency matrix of a graph $G$, we can decide whether $G$ is
$\varepsilon$-close to being bipartite or $G$ is at least
$(2+\Omega(1))\varepsilon$-far from being bipartite, by performing
$\widetilde{\mathcal{O}}\left(\frac{1}{\varepsilon ^3}\right)$ queries and in
$2^{\widetilde{\mathcal{O}}(1/\varepsilon)}$ time. This improves upon the
state-of-the-art query and time complexities of this problem of
$\widetilde{\mathcal{O}}\left(\frac{1}{\varepsilon ^6}\right)$ and
$2^{\widetilde{\mathcal{O}}(1/\varepsilon^2)}$, respectively, from the work of
Alon, Fernandez de la Vega, Kannan and Karpinski (STOC'02 and JCSS'03), where
$\widetilde{\mathcal{O}}(\cdot)$ hides a factor polynomial in $\log
\frac{1}{\varepsilon}$.