Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents

Michaela Borzechowski; John Fearnley; Spencer Gordon; Rahul Savani; Patrick Schnider; Simon Weber

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arXiv · 2402.07683

Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents

Michaela Borzechowski·John Fearnley·Spencer Gordon·Rahul Savani·Patrick Schnider·Simon Weber

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Abstract

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.

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Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, Simon Weber. 2024-02-12. Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents. https://arxiv.org/abs/2402.07683

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