Tolerant Bipartiteness Testing in Dense Graphs

TL;DR

This paper introduces a more efficient algorithm for tolerant bipartiteness testing in dense graphs, significantly reducing query complexity and runtime compared to previous methods.

Contribution

It presents the first sub-polynomial query and exponential time algorithms for tolerant bipartiteness testing in dense graphs, improving upon prior bounds.

Findings

Query complexity reduced to rac{1}{\u03b5}^3

Runtime improved to 2^{\u00f5(1/\u03b5)}

Achieves near-optimal bounds for dense graph testing

Abstract

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 $\log \frac{1}{\varepsilon}$.