The complexity of testing all properties of planar graphs, and the role of isomorphism
Abstract
Consider property testing on bounded degree graphs and let $\varepsilon>0$ denote the proximity parameter. A remarkable theorem of NewmanSohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on $\varepsilon$. Recent advances in testing minorfreeness have proven that all additive and monotone properties of planar graphs can be tested in $poly(\varepsilon^{1})$ queries. Some properties falling outside this class, such as Hamiltonicity, also have a similar complexity for planar graphs. Motivated by these results, we ask: can all properties of planar graphs can be tested in $poly(\varepsilon^{1})$ queries? Is there a uniform query complexity upper bound for all planar properties, and what is the "hardest" such property to test? We discover a surprisingly clean and optimal answer. Any property of bounded degree planar graphs can be tested in $\exp(O(\varepsilon^{2}))$ queries. Moreover, there is a matching lower bound, up to constant factors in the exponent. The natural property of testing isomorphism to a fixed graph needs $\exp(\Omega(\varepsilon^{2}))$ queries, thereby showing that (up to polynomial dependencies) isomorphism to an explicit fixed graph is the hardest property of planar graphs. The upper bound is a straightforward adapation of the NewmanSohler analysis that tracks dependencies on $\varepsilon$ carefully. The main technical contribution is the lower bound construction, which is achieved by a special family of planar graphs that are all mutually far from each other. We can also apply our techniques to get analogous results for bounded treewidth graphs. We prove that all properties of bounded treewidth graphs can be tested in $\exp(O(\varepsilon^{1}\log \varepsilon^{1}))$ queries. Moreover, testing isomorphism to a fixed forest requires $\exp(\Omega(\varepsilon^{1}))$ queries.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.10547
 Bibcode:
 2021arXiv210810547B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity