[Paper Review] Kernel(s) for Problems With no Kernel: On Out-Trees With Many Leaves

Henning Fernau, Fedor V. Fomin+4 more|ArXiv.org|Oct 27, 2008

Advanced Graph Theory Research13 references4 citations

TL;DR

This paper establishes the first polynomial kernel for Rooted $k$-Leaf Out-Branching with $O(k^3)$ size using extremal combinatorics, while proving $k$-Leaf Out-Branching has no polynomial kernel unless the polynomial hierarchy collapses. It introduces a novel separation between many-to-one and Turing kernelization by showing $n$ independent $O(k^3)$ kernels exist for the intractable 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 %$PH=Σ_p^3$ 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}.

Motivation & Objective

• To resolve the kernelization complexity of $k$-Leaf Out-Branching, a key problem in directed graph parameterized algorithms.
• To investigate whether problems without polynomial kernels can still admit efficient data reduction via multiple independent kernels.
• To establish a separation between many-to-one kernelization and Turing kernelization in parameterized complexity.
• To provide a framework for 'cheat kernelization' where $n$ independent polynomial kernels replace a single large kernel.

Proposed method

• Develops a polynomial kernel of size $O(k^3)$ for Rooted $k$-Leaf Out-Branching using extremal combinatorics and reduction rules.
• Applies a non-trivial reduction from $k$-Leaf Out-Branching to $k$-Leaf Out-Tree by testing all possible roots.
• Uses a construction based on 'nice willow graphs' to simulate out-branchings and reduce to a single instance with $b_{\text{max}}$ leaves.
• Employs a cycle-based construction to combine $n$ independent instances of Rooted $k$-Leaf Out-Tree into a single $k$-Leaf Out-Branching instance.
• Applies the lower bound framework of Bodlaender et al. [5] to show that $k$-Leaf Out-Branching cannot have a polynomial kernel unless $PH = \Sigma_p^3$.
• Reduces the kernelized instance back to $k$-Leaf Out-Tree to derive a contradiction if a polynomial kernel existed for $k$-Leaf Out-Branching.

Experimental results

Research questions

• RQ1 Can a problem without a polynomial kernel still admit $n$ independent polynomial kernels?
• RQ2 Is there a separation between many-to-one kernelization and Turing kernelization in parameterized complexity?
• RQ3 Does the existence of multiple small kernels for a hard problem imply practical tractability despite theoretical intractability?
• RQ4 Can the framework of Bodlaender et al. [5] be extended to rule out $|I|^{O(1)}$ independent polynomial kernels?
• RQ5 Are there other natural problems that exhibit this 'cheat kernelization' behavior?

Key findings

• Rooted $k$-Leaf Out-Branching admits a polynomial kernel of size $O(k^3)$, established via extremal combinatorics and reduction rules.
• $k$-Leaf Out-Branching has no polynomial kernel unless the polynomial hierarchy collapses to $\Sigma_p^3$, as shown by a reduction from $k$-Leaf Out-Tree.
• The $n$ independent $O(k^3)$ kernels for Rooted $k$-Leaf Out-Branching provide a form of Turing kernelization, enabling efficient pre-processing for the intractable $k$-Leaf Out-Branching problem.
• The construction of a single $k$-Leaf Out-Branching instance from $n$ rooted instances using a cycle of willow graphs ensures correctness and preserves the leaf count across components.
• The paper answers affirmatively an open problem on 'cheat kernelization' by demonstrating that $n$ independent polynomial kernels can circumvent lower bounds for many-to-one kernelization.
• The result establishes a strict separation between many-to-one and Turing kernelization, showing the latter can be strictly more powerful for certain problems.