  We provide polynomial-time reductions between three search problems from
three distinct areas: the P-matrix linear complementarity problem (P-LCP),
finding the sink of a unique sink orientation (USO), and a variant of the
$\alpha$-Ham Sandwich problem. For all three settings, we show that "two
choices are enough", meaning that the general non-binary version of the problem
can be reduced in polynomial time to the binary version. This specifically
means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are
equivalent to cube USOs. These results are obtained by showing that both the
P-LCP and our $\alpha$-Ham Sandwich variant are equivalent to a new problem we
introduce, P-Lin-Bellman. This problem can be seen as a new tool for
formulating problems as P-LCPs.
