40 research outputs found
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
An approximation algorithm for feedback vertex sets in tournaments
We obtain a necessary and sufficient condition in terms of forbidden structures for tournaments to possess the min-max relation on packing and covering directed cycles, together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem in this class of tournaments. Applying the local ratio technique of Bar-Yehuda and Even to the forbidden structures, we find a 2.5-approximation polynomial time algorithm for the feedback vertex set problem in any tournament.published_or_final_versio
A 7/3-Approximation for Feedback Vertex Sets in Tournaments
We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001]
A 7/3-approximation for feedback vertex sets in tournaments
We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001]
A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem
We give the first -approximation algorithm for the cluster vertex deletion
problem. This is tight, since approximating the problem within any constant
factor smaller than is UGC-hard. Our algorithm combines the previous
approaches, based on the local ratio technique and the management of true
twins, with a novel construction of a 'good' cost function on the vertices at
distance at most from any vertex of the input graph.
As an additional contribution, we also study cluster vertex deletion from the
polyhedral perspective, where we prove almost matching upper and lower bounds
on how well linear programming relaxations can approximate the problem.Comment: 23 pages, 3 figure