TY - GEN
T1 - Subquadratic kernels for implicit 3-hitting set and 3-set packing problems
AU - Le, Tien Nam
AU - Lokshtanov, Daniel
AU - Saurabh, Saket
AU - Thomassé, Stéphan
AU - Zehavi, Meirav
N1 - Publisher Copyright:
© Copyright 2018 by SIAM.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P3Packing. For these four problems kernels with O(k2) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of sets of size at most 3 (3-Hitting Set). In this paper, we give the first kernels for FVST, CVD, TPT and Induced P3-Packing with a subquadratic number of vertices. Specifically, we obtain the following results. FVST admits a kernel with O(k 32 ) vertices. CVD admits a kernel with O(k 53 ) vertices. TPT admits a kernel with O(k 32 ) vertices. Induced P3-Packing admits a kernel with O(k 53 ) vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k2) vertices for FVST and CVD. All of our results are based on novel uses of old and new "expansion lemmas", and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all), (ii) almost none of the crown is used by the solution (as opposed to none), and (iii) if H is removed from G, then there is almost no interaction between the head and the rest (as opposed to no interaction at all).
AB - We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P3Packing. For these four problems kernels with O(k2) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of sets of size at most 3 (3-Hitting Set). In this paper, we give the first kernels for FVST, CVD, TPT and Induced P3-Packing with a subquadratic number of vertices. Specifically, we obtain the following results. FVST admits a kernel with O(k 32 ) vertices. CVD admits a kernel with O(k 53 ) vertices. TPT admits a kernel with O(k 32 ) vertices. Induced P3-Packing admits a kernel with O(k 53 ) vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k2) vertices for FVST and CVD. All of our results are based on novel uses of old and new "expansion lemmas", and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all), (ii) almost none of the crown is used by the solution (as opposed to none), and (iii) if H is removed from G, then there is almost no interaction between the head and the rest (as opposed to no interaction at all).
UR - http://www.scopus.com/inward/record.url?scp=85045557552&partnerID=8YFLogxK
U2 - 10.1137/1.9781611975031.23
DO - 10.1137/1.9781611975031.23
M3 - Conference contribution
AN - SCOPUS:85045557552
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 331
EP - 342
BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
A2 - Czumaj, Artur
PB - Association for Computing Machinery
T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
Y2 - 7 January 2018 through 10 January 2018
ER -