Merge pull request #2 from fmcinern/patch-1

Update 2024-06-30-chalopin24a.md

_posts/2024-06-30-chalopin24a.md

@@ -8,23 +8,25 @@ abstract: 'Recently, Kirkpatrick et al. [ALT 2019] and  Fallat et al. [JMLR 20
   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 \textsc{B-NCTD$^+$}
-  for NCTD$^+$ is \textsf{NP}-complete in split, co-bipartite, and bipartite graphs.
-  Surprisingly, we even prove that, unless the \textsf{ETH} fails, \textsc{B-NCTD$^+$}
+  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 $G$. These are extremely rare
-  results: it is only the second (fourth, resp.) problem in \textsf{NP} to admit such
-  a double-exponential lower bound parameterized by $\mathtt{vc}$ (treewidth, resp.),
-  and only one of very few problems to admit such an \textsf{ETH}-based conditional
-  lower bound on the number of vertices in a kernel. We complement these lower bounds
-  with matching upper bounds. 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.
-  For Gromov-hyperbolic graphs, we design an approximate NCTM$^+$ for $\mathcal{B}(G)$
+  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.'
 layout: inproceedings