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A generalization and proof of the Aanderaa-Rosenberg conjecture

Authors:

Ronald L. Rivest,

Jean VuilleminAuthors Info & Claims

STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

Pages 6 - 11

https://doi.org/10.1145/800116.803747

Published: 05 May 1975 Publication History

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Abstract

We investigate the maximum number C(P) of arguments of P that must be tested in order to compute P, a Boolean function of d Boolean arguments. We present evidence for the general conjecture that C(P)=d whenever P(0^d) @@@@ P(1^d) and P is left invariant by a transitive permutation group acting on the arguments. A non-constructive argument (not based on the construction of an “oracle”) proves the generalized conjecture for d a prime power. We use this result to prove the Aanderaa-Rosenberg conjecture by showing that at least v²/9 entries of the adjacency matrix of a v-vertex undirected graph G must be examined in the worst case to determine if G has any given non-trivial monotone graph property.

References

[1]

M.R. Best, P. van Emde Boas and H.W. Lenstra, Jr., "A Sharpened Version of the Aanderaa-Rosenberg Conjecture," (preprints from the authors) (1974).

Google Scholar

[2]

F. Harary, Graph Theory, Addison-Wesley (1969).

Google Scholar

[3]

R.C. Holt and E.M. Reingold, "On the Time Required to Detect Cycles and Connectivity in Graphs," Math. Systems Theory 6 (1972).

Google Scholar

[4]

J. Hopcroft and R. Tarjan, "Efficient Planarity Testing," Cornell University Computer Science Tech. Report TR 73-165 (1973).

Google Scholar

[5]

D. Kirkpatrick, "Determining Graph Properties from Matrix Representations," Proc. 6th SIGACT Conf., Seattle (1974).

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Google Scholar

[6]

R.J. Lipton and L. Snyder, "On the Aanderaa-Rosenberg Conjecture," SIGACT News 6 (1974).

Google Scholar

[7]

E.C. Milner and D.J.A. Welsh, "On the Computational Complexity of Graph Theoretical Properties," University of Calgary, Dept. of Mathematics, Research Paper No. 232 (1974).

Google Scholar

[8]

R.L. Rivest and J. Vuillemin, "On the Number of Argument Evaluations Required to Compute Boolean Functions," U.C. Berkeley Electronics Research Laboratory Memorandum ERL-M472 (Oct. 1974).

Google Scholar

[9]

R.L. Rivest and J. Vuillemin, "On the Time Required to Recognize Properties of Graphs from Their Adjacency Matrices," U.C. Berkeley Electronics Research Laboratory Memorandum ERL-M476 (Nov. 1974).

Google Scholar

[10]

A.L. Rosenberg, "On the Time Required to Recognize Properties of Graphs: A Problem," SIGACT News 5 (1973).

Digital Library

Google Scholar

[11]

R. Tarjan, "Depth-first Search and Linear Graph Algorithms," SIAM J. on Computing, vol. 1, no. 2 (1972).

Digital Library

Google Scholar

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Csernák TSoukup L(2023)Elusive properties of infinite graphsJournal of Graph Theory10.1002/jgt.23042105:3(427-450)Online publication date: 26-Oct-2023
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Blikstad JVan Den Brand JEfron YMukhopadhyay SNanongkai D(2022)Nearly Optimal Communication and Query Complexity of Bipartite Matching2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00113(1174-1185)Online publication date: Oct-2022
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A generalization and proof of the Aanderaa-Rosenberg conjecture

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STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

May 1975

265 pages

ISBN:9781450374194

DOI:10.1145/800116

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 05 May 1975

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Blikstad JVan Den Brand JEfron YMukhopadhyay SNanongkai D(2022)Nearly Optimal Communication and Query Complexity of Bipartite Matching2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00113(1174-1185)Online publication date: Oct-2022
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Proof of a conjecture on irredundance perfect graphs

A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free

On the Aanderaa-Rosenberg Conjecture

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