(Arcdisjoint) cycle packing in tournament: classical and parameterized complexity
Abstract
Given a tournament $T$, the problem MaxCT consists of finding a maximum (arcdisjoint) cycle packing of $T$. In the same way, MaxTT corresponds to the specific case where the collection of cycles are triangles (i.e. directed 3cycles). Although MaxCT can be seen as the LP dual of minimum feedback arc set in tournaments which have been widely studied, surprisingly no algorithmic results seem to exist concerning the former. In this paper, we prove the NPhardness of both MaxCT and MaxTT. We also show that deciding if a tournament has a cycle packing and a feedback arc set with the same size is an NPcomplete problem. In light of this, we show that MaxTT admits a vertex linearkernel when parameterized with the size of the solution. Finally, we provide polynomial algorithms for MaxTT and MaxCT when the tournament is sparse, that is when it admits a FAS which is a matching.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.06669
 Bibcode:
 2018arXiv180206669B
 Keywords:

 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics;
 F.2.2;
 G.2.2
 EPrint:
 17 pages, 2 figures