639 research outputs found
A min-max theorem on tournaments
We present a structural characterization of all tournaments T = (V, A) such that, for any nonnegative integral weight function defined on V, the maximum size of a feedback vertex set packing is equal to the minimum weight of a triangle in T. We also answer a question of Frank by showing that it is N P-complete to decide whether the vertex set of a given tournament can be partitioned into two feedback vertex sets. In addition, we give exact and approximation algorithms for the feedback vertex set packing problem on tournaments. ©2007 Society for Industrial and Applied Mathematics.published_or_final_versio
Feedback Vertex Sets in Tournaments
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
Kernels for Feedback Arc Set In Tournaments
A tournament T=(V,A) is a directed graph in which there is exactly one arc
between every pair of distinct vertices. Given a digraph on n vertices and an
integer parameter k, the Feedback Arc Set problem asks whether the given
digraph has a set of k arcs whose removal results in an acyclic digraph. The
Feedback Arc Set problem restricted to tournaments is known as the k-Feedback
Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear
vertex kernel for k-FAST. That is, we give a polynomial time algorithm which
given an input instance T to k-FAST obtains an equivalent instance T' on O(k)
vertices. In fact, given any fixed e>0, the kernelized instance has at most
(2+e)k vertices. Our result improves the previous known bound of O(k^2) on the
kernel size for k-FAST. Our kernelization algorithm solves the problem on a
subclass of tournaments in polynomial time and uses a known polynomial time
approximation scheme for k-FAST
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