Sample Compression Schemes for Balls in Graphs

Jérémie Chalopin (1) , Victor Chepoi (2) , Fionn Mc Inerney (3) , Sébastien Ratel (2) , Yann Vaxès (2)

1 DALGO - Algorithmique Distribuée
2 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle
3 CISPA - Helmholtz Center for Information Security [Saarbrücken]

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 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 $\delta$-hyperbolic graphs.

Mots clés: VC-dimension; Sample Compression Schemes; Proper Sample Compression Schemes

Domaines: Mathématique discrète [cs.DM]

Jérémie Chalopin, Victor Chepoi, Fionn Mc Inerney, Sébastien Ratel, Yann Vaxès. Sample Compression Schemes for Balls in Graphs. [Research Report] CISPA Helmholtz Center for Information Security, Saarbrücken, Germany; Aix-Marseile Université, France. 2022. ⟨hal-03705798v1⟩