Mika Göös · Juho Hirvonen · Jukka Suomela

Lower bounds for local approximation

PODC 2012 · 31st Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, Madeira, Portugal, July 2012 · doi:10.1145/2332432.2332465

authors’ version publisher’s version arXiv.org

Abstract

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient.

Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks.

As a corollary of our result and by prior work, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem.

Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α < 1 and any r, k, and g.

Publication

Darek Kowalski and Alessandro Panconesi (Eds.): PODC’12, Proceedings of the 2012 ACM Symposium on Principles of Distributed Computing, July 16–18, 2012, Madeira, Portugal, pages 175–184, ACM Press, New York, 2012

ISBN 978-1-4503-1450-3

Links

Journal Version

© ACM 2012 — This is the authors’ version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Proc. 31st Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing. http://doi.acm.org/10.1145/2332432.2332465

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author’s copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.