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.

翻译：我们提供了三个不同领域搜索问题之间的多项式时间归约：P-矩阵线性互补问题（P-LCP）、寻找唯一汇方向（USO）的汇点，以及α-火腿三明治问题的一个变体。对于所有这三种情形，我们证明了“两个选择就够了”，即问题的通用非二进制版本可在多项式时间内归约至二进制版本。这特别意味着广义P-LCP等价于P-LCP，网格USO等价于立方体USO。这些结果通过证明P-LCP和我们的α-火腿三明治变体均等价于我们引入的新问题P-Lin-Bellman而获得。该问题可被视为将问题表述为P-LCP的新工具。