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


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.


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


Journal Version

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