Feedback Vertex Sets in Tournaments

Abstract

We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs.On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740 n minimal feedback vertex sets and that there is an infinite family of tournaments, all having at least 1.5448 n minimal feedback vertex sets. This improves and extends the bounds of Moon (1971).On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal feedback vertex sets of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum size feedback vertex set in a tournament.

Improved bounds for minimal feedback vertex sets in tournaments

“…Our proof of Theorem 1 is inspired by the one of Gaspers and Mnich [6] for their weaker upper bound. Their proof works by induction on the number of nodes in the input tournament .…”

“…First, our proof of Theorem 1 is algorithmic, and shows that all minimal FVS of any tournament of order can be listed in time (1.5949 ). Second, using an algorithm by Gaspers and Mnich [6] to list all minimal FVS of a tournament with polynomial delay and in polynomial space, we directly obtain the following:…”

“…This was improved by Gaspers and Mnich [6] in 2010 to 1.5448 ≤ ( ) ≤ 1.6740 . Very recently, an improvement on the upper bound was made by Fomin et al [5], who show that ( ) ≤ 1.6667 .…”

“…In other words, a tournament is a digraph with exactly one arc between any two of its vertices. Various approaches have been suggested to solve the MINIMUM FVS problem on tournaments, including approximation algorithms [3,11], fixed-parameter algorithms [4,9] as well asalgorithm [6,13]. The running time of this approach is within a polynomial factor of the number ( ) of minimal FVS in .…”

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

“…For most of these algorithms, an upper bound on the number of enumerated subsets follows from the running time of the algorithm. Examples of such recent results, both on general graphs and on some graph classes, concern the enumeration and maximum number of minimal dominating sets, minimal feedback vertex sets, minimal subset feedback vertex sets, minimal separators, maximal induced matchings, and potential maximal cliques [2,5,6,10,13,15,17,18].…”