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Dividing sums of cycles in the semiring of functional digraphs

Florian Bridoux, Christophe Crespelle, Thi Ha Duong Phan, Adrien Richard

TL;DR

This work studies the division problem in the semiring of functional digraphs, focusing on counting sums of cycles X such that $AX=B$ for given $A$ and $B$. It develops a divide-and-conquer framework built around a decomposition lemma that uses split and reduction, reducing instances to basic compact and consistent subproblems. The main result yields a counting algorithm with time complexity $O\left(|B|^3\left(\frac{|B|}{|A|}\right)^{\mathrm{div}(\mathrm{lcm} L(A))}\right)$ that is polynomial in $|B|$ when $A$ is fixed, and introduces the principal support to further refine the search. The paper also discusses limitations, open questions, and special-case polynomial-time scenarios, providing a foundation for further progress on the general division problem in the semiring of functional digraphs.

Abstract

Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a semiring with an interesting multiplicative structure. For instance, we do not know if the following division problem can be solved in polynomial time: given two functional digraphs $A$ and $B$, does $A$ divide $B$? That $A$ divides $B$ means that there exists a functional digraph $X$ such that $AX$ is isomorphic to $B$, and many such $X$ can exist. We can thus ask for the number of solutions $X$. In this paper, we focus on the case where $B$ is a sum of cycles (a disjoint union of cycles, corresponding to the limit behavior of finite discrete-time dynamical systems). There is then a naïve sub-exponential algorithm to compute the non-isomorphic solutions $X$, and our main result is an improvement of this algorithm which has the property to be polynomial when $A$ is fixed. It uses a divide-and-conquer technique that should be useful for further developments on the division problem.

Dividing sums of cycles in the semiring of functional digraphs

TL;DR

This work studies the division problem in the semiring of functional digraphs, focusing on counting sums of cycles X such that for given and . It develops a divide-and-conquer framework built around a decomposition lemma that uses split and reduction, reducing instances to basic compact and consistent subproblems. The main result yields a counting algorithm with time complexity that is polynomial in when is fixed, and introduces the principal support to further refine the search. The paper also discusses limitations, open questions, and special-case polynomial-time scenarios, providing a foundation for further progress on the general division problem in the semiring of functional digraphs.

Abstract

Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a semiring with an interesting multiplicative structure. For instance, we do not know if the following division problem can be solved in polynomial time: given two functional digraphs and , does divide ? That divides means that there exists a functional digraph such that is isomorphic to , and many such can exist. We can thus ask for the number of solutions . In this paper, we focus on the case where is a sum of cycles (a disjoint union of cycles, corresponding to the limit behavior of finite discrete-time dynamical systems). There is then a naïve sub-exponential algorithm to compute the non-isomorphic solutions , and our main result is an improvement of this algorithm which has the property to be polynomial when is fixed. It uses a divide-and-conquer technique that should be useful for further developments on the division problem.

Paper Structure

This paper contains 17 sections, 21 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries
  4. Integers
  5. Sums of cycles
  6. Integer partitions
  7. Brute force approach on the support
  8. Decomposition lemma
  9. Split
  10. Reduction
  11. Proof of the decomposition lemma
  12. Principal support
  13. Concluding remarks
  14. Acknowledgments
  15. Irreducible and prime sums of cycles
  16. ...and 2 more sections

Key Result

Theorem 1

There is an algorithm that, given two non-empty sums of cycles $A,B$, computes the number of sums of cycles $X$ satisfying $AX=B$ with time complexity in

Figures (3)

  • Figure 1: A functional digraph, with cyclic part in red, and transient part in green.
  • Figure 2: Product of two functional digraphs.
  • Figure 3: The two irreducible factorizations of $C_2+C_2$.

Theorems & Definitions (49)

  • Theorem 1
  • Definition 1
  • Example 1
  • Lemma 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 2: Decomposition lemma
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:main']} assuming Lemma \ref{['lem:main']}.
  • ...and 39 more