The Complexity of Recognizing Unique Sink Orientations

Authors Bernd Gärtner, Antonis Thomas

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Bernd Gärtner
Antonis Thomas

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Bernd Gärtner and Antonis Thomas. The Complexity of Recognizing Unique Sink Orientations. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 341-353, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Given a Boolean Circuit with n inputs and n outputs, we want to decide if it represents a Unique Sink Orientation (USO). USOs are useful combinatorial objects that serve as abstraction of many relevant optimization problems. We prove that recognizing a USO is coNP-complete. However, the situation appears to be more complicated for recognizing acyclic USOs. Firstly, we give a construction to prove that there exist cyclic USOs where the smallest cycle is of superpolynomial size. This implies that the straightforward representation of a cycle (i.e. by a list of vertices) does not make up for a coNP certificate. Inspired by this fact, we investigate the connection of recognizing an acyclic USO to PSPACE and we prove that the problem is PSPACE-complete.
  • complexity
  • recognizing
  • unique sink orientations
  • coNP


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