A fast solution to the conjugacy problem in the four-strand braid group

We present an algorithm for solving the conjugacy search problem in the fourstrand braid group. The computational complexity is cubic with respect to the braid length.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo RegionalFondo Nacional de Desarrollo Científico y Tecnológico (Chile)Center of Dynamical Systems and Related Field

Conjugacy in Garside groups I: Cyclings, powers, and rigidity

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X^m is conjugate to a rigid element, then m can be bounded above by ||\Delta||^3. In the particular case of braid groups, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors' program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.Comment: 41 page

Properties of graphs with large girth

This thesis is devoted to the analysis of a class of iterative probabilistic algorithms in regular graphs, called locally greedy algorithms, which will provide bounds for graph functions in regular graphs with large girth. This class is useful because, by conveniently setting the parameters associated with it, we may derive algorithms for some well-known graph problems, such as algorithms to find a large independent set, a large induced forest, or even a small dominating set in an input graph G. The name ``locally greedy" comes from the fact that, in an algorithm of this class, the probability associated with the random selection of a vertex v is determined by the current state of the vertices within some fixed distance of v. Given r > 2 and an r-regular graph G, we determine the expected performance of a locally greedy algorithm in G, depending on the girth g of the input and on the degree r of its vertices. When the girth of the graph is sufficiently large, this analysis leads to new lower bounds on the independence number of G and on the maximum number of vertices in an induced forest in G, which, in both cases, improve the bounds previously known. It also implies bounds on the same functions in graphs with large girth and maximum degree r and in random regular graphs. As a matter of fact, the asymptotic lower bounds on the cardinality of a maximum induced forest in a random regular graph improve earlier bounds, while, for independent sets, our bounds coincide with asymptotic lower bounds first obtained by Wormald. Our result provides an alternative proof of these bounds which avoids sharp concentration arguments. The main contribution of this work lies in the method presented rather than in these particular new bounds. This method allows us, in some sense, to directly analyse prioritised algorithms in regular graphs, so that the class of locally greedy algorithms, or slight modifications thereof, may be applied to a wider range of problems in regular graphs with large girth

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres