Février 2023 — Algorithmique et Structures Discrètes

Speaker : Bernadette Charron-Bost ( DIENS )
Title : Distributed Computation of the Average.
Abstract: Computing the average of initial values in a multi-agent system is a fundamental problem in distributed control theory. Unfortunately, an impossibility result by Boldi and Vigna shows that deterministic, anonymous agents communicating by broadcast cannot compute functions dependent on the multiplicity of the input values. In particular, they cannot compute the average of their initial values.
I will first recall the proof developed by Boldi and Vigna using the theory of graph fibrations, and will then review various approaches for circumventing the impossibility result of computing the average. I will notably present a randomized algorithm that computes the average in linear time in the size $n$ of the network, but only with high probability. I will also present a deterministic algorithm, inspired by the popular Metropolis-Hasting method in statistical physics, that computes the average in $O(n^4)$ rounds, under the additional assumption of bidirectional links.
(Joint work with Patrick Lambein-Monette.)

Speaker : Oscar Defrain (LIS – ACRO team)
Title : Minimal dominating sets enumeration in bipartite graphs.
Abstract : Minimal dominating sets enumeration has gained considerable attention since it has been shown equivalent to the long-standing open problem of dualizing a pair of monotone Boolean functions, i.e., deciding whether two positive CNF and DNF are equal. This result, obtained by Kanté, Limouzy, Mary and Nourine in 2014, holds even in the very restricted case of co-bipartite graphs (the vertex set of the graph may be partitioned into two cliques), and was strengthened by a tractable result on split graphs (the vertex set may be partitioned into a clique and an independent set). This has left the case of bipartite graphs (the vertex set may be partitioned into two independent sets) at the center of the attention in the subsequent years. We show it to be tractable inspired by a technique by Eiter, Gottlob and Makino from 2003.
(This is a joint work with Marthe Bonamy, Marc Heinrich, Michał Pilipczuk, and Jean-Florent Raymond.)

Speaker : Pablo Concha Vega (LIS – CANA team)
Title : On the Complexity of Stable and Biased Majority.
Abstract: A majority automata is a two-state cellular automata, where each cell updates its state according to the most represented state in its neighborhood. A question that naturally arises in the study of these dynamical systems asks whether there exists an efficient algorithm that can be implemented in order to compute the state configuration reached by the system at a given time-step. This problem is called the prediction problem. In this work, we study the prediction problem for a more general setting in which the local functions can be different according to their behavior in tie cases. We define two types of local rules: the stable majority and biased majority. The first one remains invariant in tie cases, and the second one takes the value 1. We call this class the heterogeneous majority cellular automata (HMCA). For this latter class, we prove that in two or more dimensions, the problem is P-Complete as a consequence of the capability of the system of simulating Boolean circuits.