Finding an Unknown Acyclic Orientation of a Given Graph

TL;DR

This paper investigates the minimum number of edge tests needed to determine an unknown acyclic orientation of a graph, providing upper bounds and NP-hardness results for approximation.

Contribution

It proves an upper bound of (1/4+o(1))n^2 on c(G) for any n-vertex graph and establishes NP-hardness of approximating c(G) within a specific factor.

Findings

c(G) q (1/4+o(1))n^2 for any graph G on n vertices

Determined NP-hardness of approximating c(G) within a factor of 74/73-e

Answered an open question by Aigner, Triesch, and Tuza

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)\le (1/4+o(1))n^2 for any graph G on n vertices, answering in the affirmative a question of Aigner, Triesch, and Tuza [Discrete Mathematics, 144 (1995) 3-10]. Also, we show that, for every e>0, it is NP-hard to approximate the parameter c(G) within a multiplicative factor 74/73-e.