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Theses

Combinatorial Algorithms and Optimization

George Manoussakis 1
1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
LRI - Laboratoire de Recherche en Informatique
Abstract : We investigate three main questions in this thesis. The first two are related to graph algorithmic problems. Given general or restricted classes of graphs, we design algorithms in order to achieve some given result. We start by introducing the class of k-degenerate graphs which are often used to model sparse real world graphs. We then focus on enumeration questions for these graphs. That is, we try and provide algorithms which must output, without duplication, all the occurrences of some input subgraph with some given properties. In the scope of this thesis, we investigate the questions of finding all subgraphs which have the property to be cycles of some given size and all subgraphs which have the property to be maximal cliques in the input sparse graph. Our two main contributions related to these problems are a worst-case output size optimal algorithm for fixed-size cycle enumeration and an output sensitive algorithm for maximal clique enumeration for this restricted class of graphs. The second main object that we study is also related to graph algorithmic questions, although in a very dierent setup. We want to consider graphs in a distributed manner. Each vertex or node has some computing power and can communicate with its neighbors. Nodes must then cooperate in order to solve a global problem. In this context, we mainly investigate questions related to finding matchings (a set of edges of the graph with no common end vertices) assuming any possible initialization (correct or incorrect) of the system. These algorithms are often referred to as self-stabilizing since no assumption is made on the initial state of the system. In this context, our two main contributions are the rst polynomial time self-stabilizing algorithm returning a 2/3-approximation of the maximum matching and a new self-stabilizing algorithm for maximal matching when communication is restricted in such a way as to simulate the message passing paradigm. Our third object of study is not related to graph algorithms, although some classical techniques are borrowed from that field to achieve some of our results. We introduce and investigate some special families of polytopes, namely primitive zonotopes, which can be described as the Minkowski sum of short primitive vectors. We prove some of their combinatorial properties and highlight connections with the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Our main contributions are new lower bounds for this diameter question as well as descriptions of small instances of these polytopes.
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https://hal.archives-ouvertes.fr/tel-01835110
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Submitted on : Monday, July 23, 2018 - 11:17:07 AM
Last modification on : Thursday, July 8, 2021 - 3:50:33 AM
Long-term archiving on: : Wednesday, October 24, 2018 - 12:19:55 PM

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George Manoussakis. Combinatorial Algorithms and Optimization. Computer Science [cs]. Paris-Sud XI, 2017. English. ⟨tel-01835110⟩

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