## Septembre 2019 — Algorithmique et Structures Discrètes

« Distributed labeling schemes on metrically defined graphs » par S. Ratel (LIS)
« Fast Algorithms for Unit Disk Graphs » par G. Dias da Fonseca (LIS)
« Two open problems of Digital Geometry with convex lattice sets » par Y. Gérard (LIS)

Orateur : Sébastien Ratel (LIS)
Titre : Distributed labeling schemes on metrically defined graphs

Orateur : Guilherme Dias da Fonseca (MCF, LIS)
Titre : Fast Algorithms for Unit Disk Graphs
Résumé : Unit-disk graphs are graphs whose $n$ vertices correspond to unit disks in the plane and whose $m$ edges correspond to pairs of intersecting disks. \emph{Graph-based} algorithms receive as input solely the adjacency representation of the graph while geometric algorithms receive the coordinates of the disks. In this talk, we consider approximation algorithms to two classic hard optimization problems: maximum independent set and minimum dominating set. We are particularly interested in algorithms whose running times are close to linear in the input size, i.e. close to $O(n+m)$ for graph-based algorithms and close to $O(n)$ for geometric algorithms. We will discuss four different approaches to obtain such algorithms: greedy, local search, strip decomposition, and shifting coresets, comparing their performance for different problems and graph classes. This work is based on multiple papers co-written by the author with Celina M. H. de Figueiredo, Vinicius G. Pereira de Sá, Raphael Machado, Gautam K. Das, and Ramesh K. Jallu.

Orateur : Yan Gerard (PR, UCA)
Titre : Two open problems of Digital Geometry with convex lattice sets
Résumé : A lattice set of $Z^d$ is digital convex if its real convex hull does not contain any other integer point than the ones of S. Problem 1: recognition of digital polyhedra. A n-digital polyhedron is the intersection of the lattice $Z^d$ with a polyhedron of $R^d$ with at most $n$ faces. Given $n$ and a convex lattice set $S$, we want to determine whether $S$ is a $n$-digital polyhedron. In dimension $d=2$, with $n=3$, the question is to determine whether there exists a triangle $T$ whose intersection with the lattice is $S$. Whatever the value of $n$, it is a very natural question related to polyhedral separability but it is still undetermined whether it is decidable in the cases where $d>3$ and $S$ is hollow (hollow means that the interior of its convex hull does not contain any integer point). Problem 2: Discrete Tomography We consider the problem of reconstruction of convex lattice sets (or HV-convex polyominoes) from their horizontal and vertical X-rays or projections. In other words, we know the number of points of a convex lattice set $S$ on each row and column, and we want to reconver $S$. It’s not known whether it can be done in polynomial time. The usual approach to tackle the problem is based on some combinatorial objects called the switching components. We present some recent results abour their structures.