Kernel(s) for Problems With no Kernel: On Out-Trees With Many Leaves

TL;DR

This paper introduces the first polynomial kernel for the rooted k-leaf out-branching problem using extremal combinatorics, and proves that no polynomial kernel exists for the general problem unless the polynomial hierarchy collapses.

Contribution

It provides the first polynomial kernel for rooted k-leaf out-branching and establishes kernelization limits for the unrooted version, distinguishing between many-to-one and Turing kernelization.

Findings

Polynomial kernel of cubic size for rooted k-leaf out-branching

No polynomial kernel for unrooted k-leaf out-branching unless PH collapses

Data reduction to multiple O(k^3) kernels for the unrooted problem

Abstract

The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms {alonLNCS4596,AlonFGKS07fsttcs,BoDo2,KnLaRo}. In this paper we step aside and take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the {\sc $k$-Leaf-Out-Branching} problem we show that no polynomial kernel is possible unless polynomial hierarchy collapses to third level %$PH=\Sigma_p^3$ by applying a recent breakthrough result by Bodlaender et al. {BDFH08} in a non-trivial fashion. However our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable the {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding "cheat kernelization" raised in {IWPECOPEN08}.