Testing Forest-Isomorphism in the Adjacency List Model

TL;DR

This paper presents a polylogarithmic query complexity algorithm for testing forest isomorphism in the adjacency list model, along with a lower bound, and demonstrates that all graph properties are testable in forests with similar efficiency.

Contribution

It introduces a new efficient property testing algorithm for forest isomorphism and establishes fundamental lower bounds, advancing understanding of graph property testing in the adjacency list model.

Findings

Polylogarithmic query complexity for forest-isomorphism testing.

Lower bound of Omega(sqrt(log n)) queries for the problem.

All graph properties are testable in forests with polylogarithmic queries.

Abstract

We consider the problem of testing if two input forests are isomorphic or are far from being so. An algorithm is called an $\varepsilon$-tester for forest-isomorphism if given an oracle access to two forests $G$ and $H$ in the adjacency list model, with high probability, accepts if $G$ and $H$ are isomorphic and rejects if we must modify at least $\varepsilon n$ edges to make $G$ isomorphic to $H$. We show an $\varepsilon$-tester for forest-isomorphism with a query complexity $\mathrm{polylog}(n)$ and a lower bound of $\Omega(\sqrt{\log{n}})$. Further, with the aid of the tester, we show that every graph property is testable in the adjacency list model with $\mathrm{polylog}(n)$ queries if the input graph is a forest.