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Finding an Unknown Acyclic Orientation of a Given Graph

Published online by Cambridge University Press:  09 July 2009

OLEG PIKHURKO
OLEG PIKHURKO*
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (web: http://www.math.cmu.edu/~pikhurko)
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Abstract

Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of comparisons needed to sort n numbers.

We prove that c(G) ≤ (1/4 + o(1))n2 for any graph G on n vertices, answering in the affirmative a question of Aigner, Triesch and Tuza [Discrete Mathematics144 (1995) 3–10]. Also, we show that, for every ϵ > 0, it is NP-hard to approximate the parameter c(G) within a multiplicative factor 74/73 − ϵ.

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Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 1 , January 2010 , pp. 121 - 131
Copyright
Copyright © Cambridge University Press 2009

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