Sample compression size versus VC dimension

Determine whether every concept class C ⊆ {0,1}^n admits a sample compression scheme whose size is polynomial in its VC dimension, and in particular whether a scheme of size O(vc(C)) exists for all such classes.

Background

Sample compression schemes aim to summarize a realizable labeled sample by a small subsample and auxiliary bits so that a reconstructor produces a hypothesis consistent with the full sample. The long-standing Sample Compression Conjecture of Floyd and Warmuth posits a scheme of size O(vc(C)) for every concept class, but only exponential-size schemes are known in general.

This paper shows barriers to one prominent approach (embedding into extremal classes) but does not resolve whether linear or polynomial-size compression schemes exist universally. The authors explicitly note the question remains open and reference Floyd (1995) and Warmuth (2003).