Theory of Computing Systems

, Volume 53, Issue 3, pp 467–502 | Cite as

The Complexity of Computing Minimal Unidirectional Covering Sets

  • Dorothea BaumeisterEmail author
  • Felix Brandt
  • Felix Fischer
  • Jan Hoffmann
  • Jörg Rothe


A common thread in the social sciences is to identify sets of alternatives that satisfy certain notions of stability according to some binary dominance relation. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer (in Math. Soc. Sci. 56(2):254–268, 2008) proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal unidirectional (i.e., either upward or downward) covering set. For both problems, we raise this lower bound to the \(\varTheta_{2}^{p}\) level of the polynomial hierarchy and provide a \(\varSigma_{2}^{p}\) upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size unidirectional covering sets are hard or complete for either of NP, coNP, and \(\varTheta_{2}^{p}\). An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer’s result that minimal bidirectional covering sets are polynomial-time computable.


Computational social choice Computational complexity Minimal upward covering sets Minimal downward covering sets 



This work was supported in part by DFG grants BR-2312/6-1, RO-1202/12-1 (within the European Science Foundation’s EUROCORES program LogICCC), BR 2312/3-2, RO-1202/11-1, and RO-1202/15-1, and by the Alexander von Humboldt Foundation’s TransCoop program. This work was done in part while the fifth author was visiting the University of Rochester.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Dorothea Baumeister
    • 1
    Email author
  • Felix Brandt
    • 2
  • Felix Fischer
    • 3
  • Jan Hoffmann
    • 4
  • Jörg Rothe
    • 1
  1. 1.Institut für InformatikHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany
  2. 2.Institut für InformatikTechnische Universität MünchenGarchingGermany
  3. 3.Statistical LaboratoryUniversity of CambridgeCambridgeUK
  4. 4.Department of Computer ScienceYale UniversityNew HavenUSA

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