Decomposition and factorisation of transients in Functional Graphs

TL;DR

This paper develops algorithms for decomposing and factorising functional graphs, extending existing methods to handle general connected graphs and providing tools to validate hypotheses about their structure.

Contribution

It introduces a polynomial-time semi-decision algorithm and an exponential-time algorithm for solving equations in general connected functional graphs, broadening the scope beyond cycle-only graphs.

Findings

The upper bound on solutions relates to cycle size in the coefficient graph.

The semi-decision algorithm efficiently constrains potential solutions.

The exponential algorithm finds all solutions with optimized heuristics.

Abstract

Functional graphs (FGs) model the graph structures used to analyse the behaviour of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be quite large, it is interesting to decompose and factorise them into several subgraphs acting together. Polynomial equations over functional graphs provide a formal way to represent this decomposition and factorisation mechanism, and solving them validates or invalidates hypotheses on their decomposability. The current solution method breaks down a single equation into a series of basic equations of the form AxX = B (with A, X, and B being FGs) to identify the possible solutions. However, it is able to consider just FGs made of cycles only. This work proposes an algorithm for solving these basic equations for general connected FGs. By exploiting a connection with the cancellation problem, we prove that the upper bound to the number of solutions is closely related to the size of the cycle in the coefficient A of the equation. The cancellation problem is also involved in the main algorithms provided by the paper. We introduce a polynomial-time semi-decision algorithm able to provide constraints that a potential solution will have to satisfy if it exists. Then, exploiting the ideas introduced in the first algorithm, we introduce a second exponential-time algorithm capable of finding all solutions by integrating several 'hacks' that try to keep the exponential as tight as possible.