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

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

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Continuous optimization
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Computational geometry
Keywords
  • P-LCP
  • Unique Sink Orientation
  • α-Ham Sandwich
  • search complexity
  • TFNP
  • UEOPL

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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 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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Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber. Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.32

Author Details

Michaela Borzechowski
  • Department of Mathematics and Computer Science, Freie Universität Berlin, Germany
John Fearnley
  • Department of Computer Science, University of Liverpool, UK
Spencer Gordon
  • Department of Computer Science, University of Liverpool, UK
Rahul Savani
  • The Alan Turing Institute, London, UK
  • and Department of Computer Science, University of Liverpool, UK
Patrick Schnider
  • Department of Computer Science, ETH Zürich, Switzerland
Simon Weber
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

  • Borzechowski, Michaela: DFG within GRK 2434 Facets of Complexity
  • Fearnley, John: EPSRC Grant EP/W014750/1
  • Gordon, Spencer: EPSRC Grant EP/W014750/1.
  • Savani, Rahul: EPSRC Grant EP/W014750/1
  • Weber, Simon: Swiss National Science Foundation under project no. 204320

Acknowledgements

We wish to thank Bernd Gärtner for introducing the authors to each other.

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