Spotting Trees with Few Leaves

TL;DR

This paper extends algorithms for detecting specific trees and paths in graphs, achieving faster runtimes and breaking previous barriers, especially in graphs with certain coloring properties.

Contribution

It generalizes existing techniques to find trees with few leaves efficiently and improves algorithms for the $k$-Internal Spanning Tree problem, surpassing the $2^n$ barrier.

Findings

New $O^*(1.657^k 2^{l/2})$ algorithm for finding $k$-vertex trees with $l$ leaves.

Improved $O^*( ext{min}(3.455^k, 1.946^n))$ algorithm for $k$-Internal Spanning Tree.

Enhanced bounds for $k$-Path and Hamiltonicity in graphs with bounded degree or specific colorings.

Abstract

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8.