Reduction from general α-Ham Sandwich to binary SWS-Colorful-Tangent

Establish whether computing (α_1, …, α_d)-cuts for arbitrary α in the α-Ham Sandwich setting can be reduced in polynomial time to SWS-Colorful-Tangent, the binary case with α_i ∈ {1, |P_i|} under strong well-separation. Either construct an explicit reduction or separate the computational complexities of these two search problems.

Background

The paper shows an equivalence between SWS-Colorful-Tangent (binary α choices) and the new P-Lin-Bellman formulation, yielding reductions to P-LCP. For arbitrary α, only existence and uniqueness (under well-separation) are guaranteed; the authors did not obtain a reduction to the binary tangent case.

Resolving whether arbitrary α-cuts reduce to the binary SWS-Colorful-Tangent would extend the “two choices are enough” theme to general α-Ham Sandwich searches, unifying geometric and algebraic formulations.