Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914803634667520 |
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| author | Borzechowski, Michaela Fearnley, John Gordon, Spencer Savani, Rahul Schnider, Patrick Weber, Simon |
| author_facet | Borzechowski, Michaela Fearnley, John Gordon, Spencer Savani, Rahul Schnider, Patrick Weber, Simon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07683 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents Borzechowski, Michaela Fearnley, John Gordon, Spencer Savani, Rahul Schnider, Patrick Weber, Simon Computational Complexity Computational Geometry Optimization and Control 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. |
| title | Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents |
| topic | Computational Complexity Computational Geometry Optimization and Control |
| url | https://arxiv.org/abs/2402.07683 |