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：我们给出了来自三个不同领域的三个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）的汇点问题，以及$\alpha$-火腿三明治问题的一个变体。对于所有这些情况，我们证明了“两个选择就足够了”，即一般非二元版本的问题可以在多项式时间内归约到二元版本。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。这些结果是通过证明P-LCP和我们的$\alpha$-火腿三明治变体都等价于我们引入的一个新问题P-Lin-Bellman而获得的。该问题可以视为将问题表述为P-LCP的一种新工具。