Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems

Abstract

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 P 3 -Packing. For these four problems kernels with O(k 2 ) 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 P 3 -Packing with a subquadratic number of vertices. Specifically, we obtain the following results.• FVST admits a kernel with O(k 3 2 ) vertices.• CVD admits a kernel with O(k 5 3 ) vertices.• TPT admits a kernel with O(k 3 2 ) vertices.• Induced P 3 -Packing admits a kernel with O(k Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k 2− ) 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).

Subset Feedback Vertex Set in Chordal and Split Graphs

“…It is natural to suspect that this lost information could have been exploited to obtain better algorithms and smaller kernels for the original problem. This was most recently vindicated by the work of Le et al [20] who designed kernels with a sub-quadratic number of vertices for several implicit 3-HS problems on graphs, improving on long-standing quadratic upper bounds in each case. Our work is in the same spirit as that of Le et al: we obtain improved results for two implicit 3-HITTING SET problems-namely: intersecting all T -triangles in chordal (respectively, split) graphs-by a careful analysis of structural properties of the input graph.…”

Packing Arc-Disjoint Cycles in Tournaments

“…MinFAST and MinFVST are NP-hard [3,13] while FAST and FVST are FPT when parameterized by the solution size k [4,23,25,31]. Further, FAST has a kernel with O(k) vertices and FVST has a kernel with O(k 1.5 ) vertices [9,35]. Surprisingly, the duals (in the linear programming sense) of MinFAST and MinFVST have not been considered in the literature until recently.…”

“…A straightforward application of the colour coding technique [5] shows that this problem is FPT and a kernel with O(k 2 ) vertices is an immediate consequence of the quadratic element kernel known for 3-Set Packing [1]. Recently, a kernel with O(k 1.5 ) vertices was shown for this problem using interesting variants and generalizations of the popular expansion lemma [35].…”

Polyhedral Properties of the Induced Cluster Subgraphs

“…Later, Hüffner et al [30] and Boral et al [9] improved this result to O(2 k k 9 +nm) and O(1.9102 k (n+m)), respectively. Le et al [35] showed that k-CVD admits a kernel with O(k 5/3 ) vertices, which means there exists a polynomial-time algorithm that converts an n-vertex instance of k-CVD to an equivalent instance with O(k 5/3 ) vertices. Fomin et al [23] used the parameterized results of [9] to show that the Minimum CVD problem (hence, Maximum IUC) is solvable in O(1.4765 n+o(n) ) time, which is better than the former result of [22].…”