 
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 NPhard under randomized reductions. Here we show that it is in fact NPhard. This
settles a conjecture of BangJensen 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. BangJensen and
Thomassen conjectured in [6] that this problem is NPhard, and Ailon, Charikar and Newman proved
in [1] that it is NPhard under randomized reductions. Here we show how to derandomize a variant
of the construction of [1] and prove that the problem is indeed NPhard. This is based on the known
fact that the minimum feedback arcset problem for general digraphs is NPhard, (see [7], p. 192),
and on certain pseudorandom properties of the quadratic residue tournaments described in [4], pp.
134137. 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
