Conflict Packing Yields Linear Vertex-Kernels for k -FAST, k -dense RTI and a Related Problem

Abstract

We develop a technique that we call Conflict Packing in the context of kernelization [7]. We illustrate this technique on several well-studied problems: Feedback Arc Set in Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a tournament T = (V,A) and seeks a set of at most k arcs whose reversal in T results in an acyclic tournament. While a linear vertex-kernel is already known for this problem [6], using the Conflict Packing allows us to find a so-called safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the Dense Rooted Triplet Inconsistency problem, one is given a set of vertices V and a dense collection $\mathcal{R}$ of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from $\mathcal{R}$. Using again the Conflict Packing, we prove that the Dense Rooted Triplet Inconsistency problem admits a linear vertex-kernel. This result improves the best known bound of O(k 2) vertices for this problem [16]. Finally, we use this technique to obtain a linear vertex-kernel for Betweenness in Tournaments, where one is given a set of vertices V and a dense collection $\mathcal{R}$ of betweenness triplets and seeks an ordering containing all but at most k triplets from $\mathcal{R}$. To the best of our knowledge this result constitutes the first polynomial kernel for the problem.

Research supported by the AGAPE project (ANR-09-BLAN-0159) and the Phylariane project (ANR-08-EMER-011-01).