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

TL;DR

This paper introduces Conflict Packing, a unifying kernelization technique that improves polynomial kernels for various dense editing problems, simplifying existing methods and achieving new linear kernels.

Contribution

The paper presents Conflict Packing, a novel unifying technique that simplifies and improves kernelization results for multiple dense graph editing problems.

Findings

Achieved a simpler kernelization for Feedback Arc Set in bipartite tournaments.

Obtained a quadratic kernel for bipartite tournament feedback arc set.

Proved linear kernels for Dense Rooted Triplet Inconsistency and Betweenness in Tournaments.

Abstract

We develop a technique that we call Conflict Packing in the context of kernelization, 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, using the Conflict Packing allows us to find a so-called safe partition, the central tool of the kernelization algorithm in, 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, using the concept of so-called bimodules. We believe however that providing an unifying technique to cope with such problems is interesting. Regarding Dense Rooted Triplet Inconsistency, 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 $\mathcal{R}$. 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 $O(k^2)$ vertices for this problem. 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 so-called betweenness triplets and seeks a linear ordering of the vertices containing all but at most $k$ triplets from $\mathcal{R}$.