Computing eulerian magnitude homology

TL;DR

This paper addresses the computational challenge of calculating Eulerian magnitude homology groups of graphs, introducing a new algorithm and analyzing its complexity and feasibility.

Contribution

It presents the first diagonal algorithm for computing these homology groups, leveraging graph substructure combinatorics and analyzing its computational complexity.

Findings

Computing the ranks is #W[1]-complete.

The diagonal BFS-based algorithm is parameterized by graph diameter.

Feasibility and future directions are discussed.

Abstract

In this paper tackle the problem of computing the ranks of certain eulerian magnitude homology groups of a graph G. First, we analyze the computational cost of our problem and prove that it is #W[1]-complete. Then we develop the first diagonal algorithm, a breadth-first-search-based algorithm parameterized by the diameter of the graph to calculate the ranks of the homology groups of interest. To do this, we leverage the close relationship between the combinatorics of the homology boundary map and the substructures appearing in the graph. We then discuss the feasibility of the presented algorithm and consider future perspectives.