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Relationship between general α-Ham Sandwich and SWS-Colorful-Tangent

Determine whether computing an (α1,…,αd)-cut for arbitrary α in well-separated point sets polynomial-time reduces to SWS-Colorful-Tangent (the binary/tangent case with α_i ∈ {1, |P_i|}), or establish a separation showing a potential complexity difference between α-Ham Sandwich and SWS-Colorful-Tangent.

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Background

The paper proves equivalence between certain binary/tangent geometric problems and algebraic/combinatorial formulations, but the status for general α remains unsettled.

Establishing such a reduction would extend the "two choices are enough" paradigm to the full α-Ham Sandwich setting or, conversely, reveal an inherent complexity gap.

References

We were unable to show that finding an α-cut for arbitrary α is not more difficult than finding it for α_i∈{1,|P_i|}. There might thus be a difference in the computational complexity of α-HS and SWS-Colorful-Tangent. Similarly, we also do not know whether α-Grid-USO (searching for a vertex with a specified number of outgoing edges per dimension) is not more difficult than Grid-USO (searching for a sink). Note that in the case of α-HS, it is at least known that the problem is contained in UEOPL. This is not known for α-Grid-USO.

— Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents (2402.07683 - Borzechowski et al., 12 Feb 2024) in Section 6 (Open Questions) — Missing reductions