arXiv subjects

Thomas Boudier

Publications and source records attributed to Thomas Boudier.

5 recordsLinked to original sources

arXiv2026

Classification of Local Optimization Problems in Directed Cycles

We present a complete classification of the distributed computational complexity of local optimization problems in directed cycles for both the deterministic and the randomized LOCAL model. We show that for any local optimization problem $\Pi$ (that can be of the form min-sum, max-sum, min-max, or max-min, for any local cost or utility function over some finite alphabet), and for any constant approximation ratio $\alpha$, the task of finding an $\alpha$-approximation of $\Pi$ in directed cycles has one of the following complexities: 1. $O(1)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 2. $\Theta(\log^* n)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 3. $\Theta(\log^* n)$ rounds in deterministic LOCAL, $\Theta(\log^* n)$ rounds in randomized LOCAL, 4. $\Theta(n)$ rounds in deterministic LOCAL, $\Theta(n)$ rounds in randomized LOCAL. Moreover, for any given $\Pi$ and $\alpha$, we can determine the complexity class automatically, with an efficient (centralized, sequential) meta-algorithm, and we can also efficiently synthesize an asymptotically optimal distributed algorithm. Before this work, similar results were only known for local search problems (e.g., locally checkable labeling problems). The family of local optimization problems is a strict generalization of local search problems, and it contains numerous commonly studied distributed tasks, such as the problems of finding approximations of the maximum independent set, minimum vertex cover, minimum dominating set, and minimum vertex coloring.

cs.DC↗

arXiv2025

Orientation does not help with 3-coloring a grid in online-LOCAL

The online-LOCAL and SLOCAL models are extensions of the LOCAL model where nodes are processed in a sequential but potentially adversarial order. So far, the only problem we know of where the global memory of the online-LOCAL model has an advantage over SLOCAL is 3-coloring bipartite graphs. Recently, Chang et al. [PODC 2024] showed that even in grids, 3-coloring requires $\Omega(\log n)$ locality in deterministic online-LOCAL. This result was subsequently extended by Akbari et al. [STOC 2025] to also hold in randomized online-LOCAL. However, both proofs heavily rely on the assumption that the algorithm does not have access to the orientation of the underlying grid. In this paper, we show how to lift this requirement and obtain the same lower bound (against either model) even when the algorithm is explicitly given a globally consistent orientation of the grid.

cs.DC↗

arXiv2025

Distributed Algorithms for Potential Problems

In this work, we present a fast distributed algorithm for local potential problems: these are graph problems where the task is to find a locally optimal solution where no node can unilaterally improve the utility in its local neighborhood by changing its own label. A simple example of such a problem is the task of finding a locally optimal cut, i.e., a cut where for each node at least half of its incident edges are cut edges. The distributed round complexity of the locally optimal cut problem has been wide open; the problem is known to require $\Omega(\log n)$ rounds in the deterministic LOCAL model and $\Omega(\log \log n)$ rounds in the randomized LOCAL model, but the only known upper bound is the trivial brute-force solution of $O(n)$ rounds. Locally optimal cut in constant-degree graphs is perhaps the simplest example of a locally checkable labeling problem for which there is still such a large gap between current upper and lower bounds. We show that in constant-degree graphs, all local potential problems, including locally optimal cut, can be solved in $\log^{O(1)} n$ rounds, both in the deterministic and randomized LOCAL models. In particular, the deterministic round complexity of the locally optimal cut problem is now settled to $\log^{\Theta(1)} n$. Our algorithms also apply to the general case of graphs of maximum degree $\Delta$. For the special case of locally optimal cut, we obtain a randomized algorithm that runs in $O(\Delta^{2} \log^{6} n)$ rounds, which can be derandomized at polylogarithmic cost with standard techniques. Furthermore, we show that a dependence in $\Delta$ is necessary: we prove a lower bound of $\Omega(\min\{\Delta,\sqrt{n}\})$ rounds, even in the quantum-LOCAL model; in particular, there is no polylogarithmic-round algorithm for the general case.

cs.DC↗

arXiv2024

Tight Lower Bounds in the Supported LOCAL Model

We study the complexity of fundamental distributed graph problems in the recently popular setting where information about the input graph is available to the nodes before the start of the computation. We focus on the most common such setting, known as the Supported LOCAL model, where the input graph (on which the studied graph problem has to be solved) is guaranteed to be a subgraph of the underlying communication network. Building on a successful lower bound technique for the LOCAL model called round elimination, we develop a framework for proving complexity lower bounds in the stronger Supported LOCAL model. Our framework reduces the task of proving a (deterministic or randomized) lower bound for a given problem $\Pi$ to the graph-theoretic task of proving non-existence of a solution to another problem $\Pi'$ (on a suitable graph) that can be derived from $\Pi$ in a mechanical manner. We use the developed framework to obtain substantial (and, in the majority of cases, asymptotically tight) Supported LOCAL lower bounds for a variety of fundamental graph problems, including maximal matching, maximal independent set, ruling sets, arbdefective colorings, and generalizations thereof. In a nutshell, for essentially any major lower bound proved in the LOCAL model in recent years, we prove a similar lower bound in the Supported LOCAL model. Our framework also gives rise to a new deterministic version of round elimination in the LOCAL model: while, previous to our work, the general round elimination technique required the use of randomness (even for obtaining deterministic lower bounds), our framework allows to obtain deterministic (and therefore via known lifting techniques also randomized) lower bounds in a purely deterministic manner. Previously, such a purely deterministic application of round elimination was only known for the specific problem of sinkless orientation [SOSA'23].

cs.DC↗

arXiv2012

Astrophysics datamining in the classroom: Exploring real data with new software tools and robotic telescopes

Within the efforts to bring frontline interactive astrophysics and astronomy to the classroom, the Hands on Universe (HOU) developed a set of exercises and platform using real data obtained by some of the most advanced ground and space observatories. The backbone of this endeavour is a new free software Web tool - Such a Lovely Software for Astronomy based on Image J (Salsa J). It is student-friendly and developed specifically for the HOU project and targets middle and high schools. It allows students to display, analyze, and explore professionally obtained astronomical images, while learning concepts on gravitational dynamics, kinematics, nuclear fusion, electromagnetism. The continuous evolving set of exercises and tutorials is being completed with real (professionally obtained) data to download and detailed tutorials. The flexibility of the Salsa J platform tool enables students and teachers to extend the exercises with their own observations. The software developed for the HOU program has been designed to be a multi-platform, multi-lingual experience for image manipulation and analysis in the classroom. Its design enables easy implementation of new facilities (extensions and plugins), minimal in-situ maintenance and flexibility for exercise plugin. Here, we describe some of the most advanced exercises about astrophysics in the classroom, addressing particular examples on gravitational dynamics, concepts currently introduced in most sciences curricula in middle and high schools.

physics.ed-ph↗