Complexity Estimates for Two Uncoupling Algorithms

Uncoupling algorithms transform a linear differential system of first order into one or several scalar differential equations. We examine two approaches to uncoupling: the cyclic-vector method (CVM) and the Danilevski-Barkatou-Z\"urcher algorithm (DBZ). We give tight size bounds on the scalar equations produced by CVM, and design a fast variant of CVM whose complexity is quasi-optimal with respect to the output size. We exhibit a strong structural link between CVM and DBZ enabling to show that, in the generic case, DBZ has polynomial complexity and that it produces a single equation, strongly related to the output of CVM. We prove that algorithm CVM is faster than DBZ by almost two orders of magnitude, and provide experimental results that validate the theoretical complexity analyses.Comment: To appear in Proceedings of ISSAC'13 (21/01/2013

Counting connected graphs with large excess

International audienceWe enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion

Enumeration and structure of inhomogeneous graphs

International audienceWe analyze a general model of weighted graphs, introduced by de Panafieu and Ravelomanana (2014) and similar to the inhomogeneous graph model of Söderberg (2002). We investigate the sum of the weights of those graphs and their structure. Those results allow us to give a new proof in a more general setting of a theorem of Wright (1972) on the enumeration of properly colored graphs. We also discuss applications related to social networksNous étudions un modèle de graphes pondérés, introduits par de Panafieu et Ravelomanana (2014) et proche des graphes inhomogènes de Söderberg (2002). Nous analysons la somme des poids de ces graphes et leur structure. Ces résultats nous permettent d’obtenir une nouvelle preuve d’un théorème de Wright (1972) sur l’énumération des graphes bien colorés, ainsi que sur un modèle lié aux réseaux sociaux