Jérémie Chalopin

Jérémie Chalopin, Maria Kokkou

A Locally Checkable Labeling (LCL) is a distributed constraint satisfaction problem defined on a bounded-degree graph that relates a finite set of input labels to a finite set of output labels through a finite set of locally checkable constraints. In this work we define labels and local constraints that encode solutions to two classical problems: leader election and spanning tree construction. It is known that leader election cannot be expressed as an LCL in arbitrary graphs using constant-size labels. In fact, it is known that there does not exist a finite set of labels and local constraints for leader election even for the class of rings. On the other hand, there exists a finite set of labels and local constraints characterizing leader election on trees. In this work, we prove that there exists a finite set of labels and local constraints for leader election also in the much larger class of dismantlable graphs. Our labels need one bit per edge or equivalently $O(Δ)$ bits per node (where $Δ$ is the maximum degree in the graph) and are checkable within the graph induced by the 1-neighborhood of each node. To the best of our knowledge, these are the first local labeling results tailored to dismantlable graphs, potentially highlighting structural properties useful for designing labels and constraints for additional LCL problems. Finally, we present a generic transformation that converts any finite set of labels and local constraints into a silent self-stabilizing algorithm by adding only one extra state, assuming a Gouda fair scheduler. This transformation may be of independent interest.

Jérémie Chalopin, Victor Chepoi, Fionn Mc Inerney et al.

One of the open problems in machine learning is whether any set-family of VC-dimension $d$ admits a sample compression scheme of size $O(d)$. In this paper, we study this problem for balls in graphs. For a ball $B=B_r(x)$ of a graph $G=(V,E)$, a realizable sample for $B$ is a signed subset $X=(X^+,X^-)$ of $V$ such that $B$ contains $X^+$ and is disjoint from $X^-$. A proper sample compression scheme of size $k$ consists of a compressor and a reconstructor. The compressor maps any realizable sample $X$ to a subsample $X'$ of size at most $k$. The reconstructor maps each such subsample $X'$ to a ball $B'$ of $G$ such that $B'$ includes $X^+$ and is disjoint from $X^-$. For balls of arbitrary radius $r$, we design proper labeled sample compression schemes of size $2$ for trees, of size $3$ for cycles, of size $4$ for interval graphs, of size $6$ for trees of cycles, and of size $22$ for cube-free median graphs. For balls of a given radius, we design proper labeled sample compression schemes of size $2$ for trees and of size $4$ for interval graphs. We also design approximate sample compression schemes of size 2 for balls of $δ$-hyperbolic graphs.

Jérémie Chalopin, Victor Chepoi, Fionn Mc Inerney et al.

Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and showed it is the most efficient machine teaching model satisfying the Goldman-Mathias collusion-avoidance criterion. A teaching map $T$ for a concept class $\mathcal{C}$ assigns a (teaching) set $T(C)$ of examples to each concept $C \in \mathcal{C}$. A teaching map is non-clashing if no pair of concepts are consistent with the union of their teaching sets. The size of a non-clashing teaching map (NCTM) $T$ is the maximum size of a teaching set $T(C)$, $C \in \mathcal{C}$. The non-clashing teaching dimension NCTD$(\mathcal{C})$ of $\mathcal{C}$ is the minimum size of an NCTM for $\mathcal{C}$. NCTM$^+$ and NCTD$^+(\mathcal{C})$ are defined analogously, except the teacher may only use positive examples. We study NCTMs and NCTM$^+$s for the concept class $\mathcal{B}(G)$ consisting of all balls of a graph $G$. We show that the associated decision problem B-NCTD$^+$ for NCTD$^+$ is NP-complete in split, co-bipartite, and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails, B-NCTD$^+$ does not admit an algorithm running in time $2^{2^{o(\text{vc})}}\cdot n^{O(1)}$, nor a kernelization algorithm outputting a kernel with $2^{o(\text{vc})}$ vertices, where vc is the vertex cover number of $G$. We complement these lower bounds with matching upper bounds. These are extremely rare results: it is only the second problem in NP to admit such a tight double-exponential lower bound parameterized by vc, and only one of very few problems to admit such an ETH-based conditional lower bound on the number of vertices in a kernel. For trees, interval graphs, cycles, and trees of cycles, we derive NCTM$^+$s or NCTMs for $\mathcal{B}(G)$ of size proportional to its VC-dimension, and for Gromov-hyperbolic graphs, we design an approximate NCTM$^+$ of size 2.

Jérémie Chalopin, Victor Chepoi, Shay Moran et al.

We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that all previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous. On the positive side we present a new construction of an unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabeled sample compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.