Recently, Kirkpatrick et al. [ALT 2019] and Fallat et al. [JMLR 2023] introduced non-clashing teaching and showed it to be the most efficient machine teaching model satisfying the benchmark for collusion-avoidance set by Goldman and Mathias. 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 $\text{NCTD}(\mathcal{C})$ of $\mathcal{C}$ is the minimum size of an NCTM for $\mathcal{C}$. $\text{NCTM}^+$ and $\text{NCTD}^+(\mathcal{C})$ are defined analogously, except the teacher may only use positive examples. We study NCTMs and $\text{NCTM}^+\text{s}$ for the concept class $\mathcal{B}(G)$ consisting of all balls of a graph $G$. We show that the associated decision problem $\text{B-NCTD}^+$ for $\text{NCTD}^+$ is NP-complete in split, co-bipartite, and bipartite graphs. Surprisingly, we even prove that, unless the ETH fails, $\text{B-NCTD}^+$ does not admit an algorithm running in time $2^{2^{o(\mathtt{vc})}}\cdot n^{\mathcal{O}(1)}$, nor a kernelization algorithm outputting a kernel with $2^{o(\mathtt{vc})}$ vertices, where $\mathtt{vc}$ 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 $\mathtt{vc}$, 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 $\text{NCTM}^+\text{s}$ or NCTMs for $\mathcal{B}(G)$ of size proportional to its VC-dimension. For Gromov-hyperbolic graphs, we design an approximate $\text{NCTM}^+$ for $\mathcal{B}(G)$ of size $2$, in which only pairs of balls with Hausdorff distance larger than some constant must satisfy the non-clashing condition.

Abstract. One of the open problems in machine learning is whether any set-family of VC-dimension [Formula: see text] admits a sample compression scheme of size [Formula: see text]. In this paper, we study this problem for balls in graphs. For a ball [Formula: see text] of a graph [Formula: see text], a realizable sample for [Formula: see text] is a signed subset [Formula: see text] of [Formula: see text] such that [Formula: see text] contains [Formula: see text] and is disjoint from [Formula: see text]. A proper sample compression scheme of size [Formula: see text] consists of a compressor and a reconstructor. The compressor maps any realizable sample [Formula: see text] to a subsample [Formula: see text] of size at most [Formula: see text]. The reconstructor maps each such subsample [Formula: see text] to a ball [Formula: see text] of [Formula: see text] such that [Formula: see text] includes [Formula: see text] and is disjoint from [Formula: see text]. For balls of arbitrary radius [Formula: see text], 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 [Formula: see text]-hyperbolic graphs.

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 balls of arbitrary radius r, we design proper sample compression schemes of size 4 for interval graphs, of size 6 for trees of cycles, and of size 22 for cube-free median graphs. We also design approximate sample compression schemes of size 2 for balls of δ-hyperbolic graphs.

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. One of the important problems is finding natural classes of graphs admitting distance labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on n nodes enjoys distance labeling scheme with labels of O(log^3 n) bits.