A Faster Exact Algorithm for the Directed Maximum Leaf Spanning Tree Problem

TL;DR

This paper introduces a faster exact algorithm for the Directed Maximum Leaf Spanning Tree problem, achieving improved exponential time bounds using a branch-and-reduce approach and Measure & Conquer analysis.

Contribution

It presents a novel branch-and-reduce algorithm with Measure & Conquer analysis that significantly improves the exponential time complexity for solving the problem.

Findings

Time complexity improved to O*(1.9043^n) with polynomial space.

Exponential space reduces runtime to O*(1.8139^n).

First such algorithms with these bounds for the problem.

Abstract

Given a directed graph $G=(V,A)$, the Directed Maximum Leaf Spanning Tree problem asks to compute a directed spanning tree (i.e., an out-branching) with as many leaves as possible. By designing a Branch-and-Reduced algorithm combined with the Measure & Conquer technique for running time analysis, we show that the problem can be solved in time $\Oh^*(1.9043^n)$ using polynomial space. Hitherto, there have been only few examples. Provided exponential space this run time upper bound can be lowered to $\Oh^*(1.8139^n)$.