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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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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.

Cite AsGet BibTex

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

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.

Subject Classification

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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