Skip to Main content Skip to Navigation
Journal articles

Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability

Maximilien Gadouleau 1 Adrien Richard 2 Eric Fanchon 3
2 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe BIOINFO
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
3 TIMC-BCM [2016-2019] - Biologie Computationnelle et Mathématique [2016-2019]
TIMC [2016-2019] - Techniques de l'Ingénierie Médicale et de la Complexité - Informatique, Mathématiques et Applications, Grenoble - UMR 5525 [2016-2019]
Abstract : Linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations towards their destinations. The solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding. The guessing number approach converts this problem to determining the number of fixed points of coding functions f : A^n → A^n over a finite alphabet A (usually referred to as Boolean networks if A = {0, 1}) with a given interaction graph, that describes which local functions depend on which variables. In this paper, we generalise the so-called reduction of coding functions in order to eliminate variables. We then determine the maximum number of fixed points of a fully reduced coding function, whose interaction graph has a loop on every vertex. Since the reduction preserves the number of fixed points, we then apply these ideas and results to obtain four main results on the linear network coding solvability problem. First, we prove that non-decreasing coding functions cannot solve any more instances than routing already does. Second, we show that triangle-free undirected graphs are linearly solvable if and only if they are solvable by routing. This is the first classification result for the linear network coding solvability problem. Third, we exhibit a new class of non-linearly solvable graphs. Fourth, we determine large classes of strictly linearly solvable graphs.
Complete list of metadatas

Cited literature [40 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01318072
Contributor : Adrien Richard <>
Submitted on : Thursday, May 19, 2016 - 11:09:50 AM
Last modification on : Friday, August 7, 2020 - 3:00:28 AM

File

1412.5310v1.pdf
Files produced by the author(s)

Identifiers

Citation

Maximilien Gadouleau, Adrien Richard, Eric Fanchon. Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2016, 62 (5), pp.2504-2519. ⟨10.1109/TIT.2016.2544344⟩. ⟨hal-01318072⟩

Share

Metrics

Record views

794

Files downloads

277