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Distributed Verification and Hardness of Distributed Approximation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sarma, Atish Das Holzer, Stephan Korman, Amos Nanongkai, Danupon Kor, Liah Pandurangan, Gopal Peleg, David Wattenhofer, Roger
Copyright Year	2012
Abstract	We study the verification problem in distributed networks, stated as follows. Let $H$ be a subgraph of a network $G$ where each vertex of $G$ knows which edges incident on it are in $H$. We would like to verify whether $H$ has some properties, e.g., if it is a tree or if it is connected (every node knows at the end of the process whether $H$ has the specified property or not). We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification and give almost tight lower bounds on the running time of distributed verification algorithms for many fundamental problems such as connectivity, spanning connected subgraph, and $s$-$t$ cut verification. We then show applications of these results in deriving strong unconditional time lower bounds on the hardness of distributed approximation for many classical optimization problems including minimum spanning tree (MST), shortest paths, and minimum cut. Many of these results are the first nontrivial lower bounds for both exact and approximate distributed computation, and they resolve previous open questions. Moreover, our unconditional lower bound of approximating MST subsumes and improves upon the previous hardness of approximation bound of Elkin [M. Elkin, SIAM J. Comput., 36 (2006), pp. 433--456] as well as the lower bound for (exact) MST computation of Peleg and Rubinovich [D. Peleg and V. Rubinovich, SIAM J. Comput., 30 (2000), pp. 1427--1442]. Our result implies that there can be no distributed approximation algorithm for MST that is significantly faster than the current exact algorithm for any approximation factor. Our lower bound proofs show an interesting connection between communication complexity and distributed computing which turns out to be useful in establishing the time complexity of exact and approximate distributed computation of many problems.
Starting Page	1235
Ending Page	1265
Page Count	31
File Format	PDF
ISSN	00975397
DOI	10.1137/11085178X
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5 (Special Section on the Forty-Third Annual ACM Symposium on Theory of Computing (STOC 2011))
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-10-25
Access Restriction	Subscribed
Subject Keyword	graph optimization problems distributed algorithms hardness of approximation Graph algorithms Network design and communication communication complexity Distributed algorithms lower bounds
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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