Unique End of Potential Line

Authors John Fearnley, Spencer Gordon, Ruta Mehta, Rahul Savani

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Author Details

John Fearnley
  • University of Liverpool, UK
Spencer Gordon
  • California Institute of Technology, Pasadena, CA, USA
Ruta Mehta
  • University of Illinois at Urbana-Champaign, IL, USA
Rahul Savani
  • University of Liverpool, UK


We thank Aviad Rubinstein and Kousha Etessami for alerting us to the work in [Qi Qi, 2012], and we thank Rasmus Ibsen Jensen for helpful comments on a preprint of this paper.

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John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique End of Potential Line. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The complexity class CLS was proposed by Daskalakis and Papadimitriou in 2011 to understand the complexity of important NP search problems that admit both path following and potential optimizing algorithms. Here we identify a subclass of CLS - called UniqueEOPL - that applies a more specific combinatorial principle that guarantees unique solutions. We show that UniqueEOPL contains several important problems such as the P-matrix Linear Complementarity Problem, finding Fixed Point of Contraction Maps, and solving Unique Sink Orientations (USOs). UniqueEOPL seems to a proper subclass of CLS and looks more likely to be the right class for the problems of interest. We identify a problem - closely related to solving contraction maps and USOs - that is complete for UniqueEOPL. Our results also give the fastest randomised algorithm for P-matrix LCP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • P-matrix linear complementarity problem
  • unique sink orientation
  • contraction map
  • TFNP
  • total search problems
  • continuous local search


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