Parameterized Algorithms for Computing MAD Trees

TL;DR

This paper analyzes the parameterized complexity of the MAD tree problem, providing efficient algorithms for certain graph classes and establishing NP-hardness in others.

Contribution

It introduces new fixed-parameter tractable algorithms for MAD trees based on graph parameters and explores the problem's complexity landscape.

Findings

Linear-time algorithm for graphs of constant modular width.

Polynomial-time algorithm for graphs with bounded treewidth.

NP-hardness on split graphs.

Abstract

We consider the well-studied problem of finding a spanning tree with minimum average distance between vertex pairs (called a MAD tree). This is a classic network design problem which is known to be NP-hard. While approximation algorithms and polynomial-time algorithms for some graph classes are known, the parameterized complexity of the problem has not been investigated so far. We start a parameterized complexity analysis with the goal of determining the border of algorithmic tractability for the MAD tree problem. To this end, we provide a linear-time algorithm for graphs of constant modular width and a polynomial-time algorithm for graphs of bounded treewidth; the degree of the polynomial depends on the treewidth. That is, the problem is in FPT with respect to modular width and in XP with respect to treewidth. Moreover, we show it is in FPT when parameterized by vertex integrity or by an above-guarantee parameter. We complement these algorithms with NP-hardness on split graphs.