A $2k$-Vertex Kernel for Maximum Internal Spanning Tree

TL;DR

This paper presents an improved kernelization technique for the maximum internal spanning tree problem, reducing the problem size to 2k vertices and enabling faster algorithms.

Contribution

It introduces a novel application of existing reduction rules combined with a greedy local exchange procedure to achieve a smaller kernel.

Findings

Achieved a 2k-vertex kernel for the problem.

Developed a deterministic algorithm with runtime 4^k * n^{O(1)}.

Enhanced the understanding of kernelization in parameterized graph problems.

Abstract

We consider the parameterized version of the maximum internal spanning tree problem, which, given an $n$-vertex graph and a parameter $k$, asks for a spanning tree with at least $k$ internal vertices. Fomin et al. [J. Comput. System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a $3k$-vertex kernel. Here we propose a novel way to use the same reduction rule, resulting in an improved $2k$-vertex kernel. Our algorithm applies first a greedy procedure consisting of a sequence of local exchange operations, which ends with a local-optimal spanning tree, and then uses this special tree to find a reducible structure. As a corollary of our kernel, we obtain a deterministic algorithm for the problem running in time $4^k \cdot n^{O(1)}$.