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

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 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 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(k2) 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 R of betweenness triplets and seeks an ordering containing all but at most k triplets from R. To the best of our knowledge this result constitutes the first polynomial kernel for the problem.