Parameterized Algorithms for Computing MAD Trees

2026-03-31

AI Summary

This study investigates the parameterized complexity of the Minimum Average Distance (MAD) spanning tree problem, which is known to be NP-hard. The work establishes the first comprehensive tractability boundaries for this problem across several structural graph parameters: it presents a linear-time algorithm on graphs of constant modular width, an XP algorithm on graphs of bounded treewidth, and proves fixed-parameter tractability (FPT) with respect to both vertex integrity and the recently introduced “surplus” parameter. Furthermore, the paper demonstrates that the MAD tree problem remains NP-hard even when restricted to split graphs. By integrating techniques from parameterized complexity theory, modular decomposition, treewidth-based dynamic programming, and surplus-parameterized algorithms, this research significantly advances the theoretical understanding and algorithmic solvability of the MAD spanning tree problem.

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.

Problem

• MAD tree
• parameterized complexity
• minimum average distance
• spanning tree
• NP-hard

Innovation

• parameterized complexity
• MAD tree
• modular width
• treewidth
• fixed-parameter tractability