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Ranking tournaments A tournament is an oriented complete graph. The feedback arc set problem for tournaments is
 

Summary: Ranking tournaments
N. Alon
Abstract
A tournament is an oriented complete graph. The feedback arc set problem for tournaments is
the optimization problem of determining the minimum possible number of edges of a given input
tournament T whose reversal makes T acyclic. Ailon, Charikar and Newman showed that this
problem is NP-hard under randomized reductions. Here we show that it is in fact NP-hard. This
settles a conjecture of Bang-Jensen and Thomassen.
1 Introduction
A tournament is an oriented complete graph. A feedback arc set in a digraph is a collection of edges
whose reversal (or removal) makes the digraph acyclic. The feedback arc set problem for tournaments
is the optimization problem of determining the minimum possible cardinality of a feedback arc set
in a given tournament. The problem for general digraphs is defined analogously. Bang-Jensen and
Thomassen conjectured in [6] that this problem is NP-hard, and Ailon, Charikar and Newman proved
in [1] that it is NP-hard under randomized reductions. Here we show how to derandomize a variant
of the construction of [1] and prove that the problem is indeed NP-hard. This is based on the known
fact that the minimum feedback arc-set problem for general digraphs is NP-hard, (see [7], p. 192),
and on certain pseudo-random properties of the quadratic residue tournaments described in [4], pp.
134-137. Similar constructions can be given using any other family of antisymmetric matrices with
{-1, 1} entries whose rows are nearly orthogonal. We note that unlike the authors of [1], we do

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics