两个选择足以解决P-LCP、USO和彩色切线问题
Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents
AI总结：
本研究提出新问题P-Lin-Bellman，证明P矩阵线性互补问题、唯一汇点定向汇点查找、α-汉堡包问题变体三者的非二元版均可多项式时间归约为二元版，为问题建模为P-LCP提供新工具。
摘要：
我们给出了三个不同领域的搜索问题之间的多项式时间归约：P矩阵线性互补问题（P-matrix linear complementarity problem，P-LCP）、寻找唯一汇点定向（unique sink orientation，USO）的汇点，以及α-汉堡包（α-Ham Sandwich）问题的一个变体。针对这三类问题，我们证明“两个选择就足够”，即问题的一般非二元版本可在多项式时间内归约为二元版本。具体而言，广义P-LCP与P-LCP等价，网格USO与立方体USO等价。我们通过证明P-LCP和所提出的α-Ham Sandwich变体均等价于我们引入的新问题P-Lin-Bellman得到上述结果，该问题可作为将问题建模为P-LCP的新工具。
英文摘要：
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 $α$-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 $α$-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.
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