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Certificate size versus dimension for non-completable partial outmaps

Investigate whether, for a fixed hypercube dimension d, non-completable partial outmaps without non-spanned or projectable dimensions exist only for certificate sizes n bounded by a function f(d), or conversely whether for a given certificate size n such certificates exist only up to dimension g(n).

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Background

The paper introduces certificates (minimal sets of assigned vertices) witnessing non-completability of partial outmaps and fully classifies 4-certificates, also showing the existence of n-certificates for n ≥ 4.

The authors pose an explicit open question about the tradeoff between certificate size and hypercube dimension, seeking structural limits on when such certificates can exist depending on d and n.

References

Moreover, an interesting open question would be whether in a fixed dimension d, there would only be n-certificates up to a small n=f(d), or if all n-certificates without non-spanned or projectable dimensions only exist in hypercubes up to d = g(n) dimensions (as this is the case for 2-certificates and 4-certificates).

— Non-Promise Version of Unique Sink Orientations (2408.17283 - Marques, 30 Aug 2024) in Section 7 (Future Work)