The $k$-Leaf Spanning Tree Problem Admits a Klam Value of 39

TL;DR

This paper presents a new parameterized algorithm for the $k$-Leaf Spanning Tree problem, improving the known klam value from 37 to 39 using a novel bounded search trees technique.

Contribution

It introduces an $O^*(3.188^k)$-time algorithm that increases the klam value for the problem, advancing the understanding of its computational complexity.

Findings

Klam value for $k$-LST increased to 39.

Algorithm achieves $O^*(3.188^k)$ runtime.

Novel application of bounded search trees technique.

Abstract

Given an undirected graph $G$ and a parameter $k$, the $k$-Leaf Spanning Tree ($k$-LST) problem asks if $G$ contains a spanning tree with at least $k$ leaves. This problem has been extensively studied over the past three decades. In 2000, Fellows et al. [FSTTCS'00] explicitly asked whether it admits a klam value of 50. A steady progress towards an affirmative answer continued until 5 years ago, when an algorithm of klam value 37 was discovered. In this paper, we present an $O^*(3.188^k)$-time parameterized algorithm for $k$-LST, which shows that the problem admits a klam value of 39. Our algorithm is based on an interesting application of the well-known bounded search trees technique, where the correctness of rules crucially depends on the history of previously applied rules in a non-standard manner.