Conflict Packing: an unifying technique to obtain polynomial kernels for editing problems on dense instances

We develop a technique that we call Conflict Packing in the context of kernelization [8], obtaining (and improving) several polynomial kernels for editing problems on dense instances. We apply this technique on several well-studied problems: Feedback Arc Set in (Bipartite) Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a (bipartite) tournament T = (V,A) and seeks a set of at most k arcs whose reversal in T results in an acyclic (bipartite) tournament. While a linear vertex-kernel is already known for the first 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. For the case of bipartite tournaments, the same technique allows us to obtain a quadratic vertex-kernel. Again, such a kernel was already known to exist [35], using the concept of so-called bimodules. We believe however that our algorithm can provide a better insight on the structural properties of the problem. Regarding Dense Rooted Triplet Inconsistency, 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. As a main consequence of our technique, we prove that the Dense Rooted Triplet Inconsistency problem admits a linear vertex-kernel. This result improves the best known bound of O(k) vertices for this problem [22]. 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 so-called betweenness triplets and seeks a linear ordering of the vertices containing all but at most k triplets from R. We would like to mention that this result has recently been improved [29], based on some of the results we present here.