Fichtenberger, H., Peng, P. orcid.org/0000-0003-2700-5699 and Sohler, C. (2019) Every testable (infinite) property of bounded-degree graphs contains an infinite hyperfinite subproperty. In: Chan, T., (ed.) Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA19). 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA19), 06-09 Jan 2019, San Diego, California. SIAM ISBN 978-1-61197-548-2
Abstract
One of the most fundamental questions in graph property testing is to characterize the combinatorial structure of properties that are testable with a constant number of queries. We work towards an answer to this question for the bounded-degree graph model introduced in [GR02], where the input graphs have maximum degree bounded by a constant d. In this model, it is known (among other results) that every hyperfinite property is constant-query testable [NS13], where, informally, a graph property is hyperfinite, if for every δ > 0 every graph in the property can be partitioned into small connected components by removing δn edges.
In this paper we show that hyperfiniteness plays a role in every testable property, i.e. we show that every testable property is either finite (which trivially implies hyperfiniteness and testability) or contains an infinite hyperfinite subproperty. A simple consequence of our result is that no infinite graph property that only consists of expander graphs is constant-query testable.
Based on the above findings, one could ask if every infinite testable non-hyperfinite property might contain an infinite family of expander (or near-expander) graphs. We show that this is not true. Motivated by our counterexample we develop a theorem that shows that we can partition the set of vertices of every bounded degree graph into a constant number of subsets and a separator set, such that the separator set is small and the distribution of k-discs on every subset of a partition class, is roughly the same as that of the partition class if the subset has small expansion.
Metadata
Item Type: | Proceedings Paper |
---|---|
Authors/Creators: |
|
Editors: |
|
Copyright, Publisher and Additional Information: | © 2019 The Authors. Reproduced in accordance with the publisher's self-archiving policy. |
Dates: |
|
Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 06 Dec 2018 11:01 |
Last Modified: | 18 Jun 2019 15:33 |
Status: | Published |
Publisher: | SIAM |
Refereed: | Yes |
Identification Number: | 10.1137/1.9781611975482.45 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:139624 |