Décomposition de graphes en plus courts chemins et en cycles de faible excentricité

Abstract : In collaboration with researchers in biology at Université Pierre et Marie Curie, we study graphs coming from biological data in order to improve our understanding of it. Those graphs come from DNA fragments, named reads. Each read is a vertex and two vertices are linked if the DNA sequences are similar enough. Such graphs have particular structure that we name hub-laminar. A graph is said to be hub-laminar if it may be represented as a (small) set of shortest paths such that every vertex of the graph is close to one of those paths. We first study the case where the graph is composed of an unique shortest path of low eccentricity. This problem was first defined by F. Dragan. We improve the proof of an approximation algorithm already existing and propose a new one, a 3 -approximation running in linear time. Furthermore we show its link with the k -laminar problem defined by Habib, consisting in finding a diameter of low eccentricity. We then define and study the problem of the isometric cycle of minimal eccentricity. We show that this problem is NP-complete and propose two approximation algorithms. We then properly define what is an hub-laminar decomposition and we show an approximation algorithm running in O(nm) time. We test this algorithm with randomly generated graphs and apply it to our biological data. Finally we show that computing an isometric cycle of low eccentricity allows to embed a graph into a cycle with a low multiplicative distortion. Computing an hub-laminar decomposition allows a compact representation of distances with a low additive distortion.
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https://tel.archives-ouvertes.fr/tel-01994139
Contributor : Léo Planche <>
Submitted on : Friday, January 25, 2019 - 12:07:44 PM
Last modification on : Friday, December 13, 2019 - 11:18:02 AM
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Léo Planche. Décomposition de graphes en plus courts chemins et en cycles de faible excentricité. Mathématique discrète [cs.DM]. Paris Descartes, 2018. Français. ⟨tel-01994139⟩

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