Sample compression schemes for balls in 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 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 \(\delta\)-hyperbolic graphs.