The Complexity of Computing Minimal Unidirectional Covering Sets

  • Dorothea Baumeister
  • Felix Brandt
  • Felix Fischer
  • Jan Hoffmann
  • Jörg Rothe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)


Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [1] 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 \(\Theta_2^p\) level of the polynomial hierarchy and provide a \(\Sigma_2^p\) 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 \(\Theta_2^p\). 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brandt, F., Fischer, F.: Computing the minimal covering set. Mathematical Social Sciences 56(2), 254–268 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Laslier, J.: Tournament Solutions and Majority Voting. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    McGarvey, D.: A theorem on the construction of voting paradoxes. Econometrica 21(4), 608–610 (1953)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fishburn, P.: Condorcet social choice functions. SIAM Journal on Applied Mathematics 33(3), 469–489 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Miller, N.: A new solution set for tournaments and majority voting: Further graph-theoretical approaches to the theory of voting. American Journal of Political Science 24(1), 68–96 (1980)CrossRefGoogle Scholar
  6. 6.
    Dutta, B.: Covering sets and a new Condorcet choice correspondence. Journal of Economic Theory 44, 63–80 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chevaleyre, Y., Endriss, U., Lang, J., Maudet, N.: A short introduction to computational social choice. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds.) SOFSEM 2007. LNCS, vol. 4362, pp. 51–69. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Woeginger, G.: Banks winners in tournaments are difficult to recognize. Social Choice and Welfare 20, 523–528 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Alon, N.: Ranking tournaments. SIAM Journal on Discrete Mathematics 20(1), 137–142 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conitzer, V.: Computing Slater rankings using similarities among candidates. In: Proceedings of the 21st National Conference on Artificial Intelligence, pp. 613–619. AAAI Press, Menlo Park (2006)Google Scholar
  11. 11.
    Brandt, F., Fischer, F., Harrenstein, P.: The computational complexity of choice sets. Mathematical Logic Quarterly 55(4), 444–459 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brandt, F., Fischer, F., Harrenstein, P., Mair, M.: A computational analysis of the tournament equilibrium set. Social Choice and Welfare (Forthcoming)Google Scholar
  13. 13.
    Wagner, K.: More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science 51, 53–80 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bordes, G.: On the possibility of reasonable consistent majoritarian choice: Some positive results. Journal of Economic Theory 31, 122–132 (1983)CrossRefzbMATHGoogle Scholar
  15. 15.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  16. 16.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  17. 17.
    Papadimitriou, C., Zachos, S.: Two remarks on the power of counting. In: Cremers, A.B., Kriegel, H.-P. (eds.) GI-TCS 1983. LNCS, vol. 145, pp. 269–276. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  18. 18.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: A richer understanding of the complexity of election systems. In: Ravi, S., Shukla, S. (eds.) Fundamental Problems in Computing: Essays in Honor of Professor Daniel J. Rosenkrantz, pp. 375–406. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Baumeister, D., Brandt, F., Fischer, F., Hoffmann, J., Rothe, J.: The complexity of computing minimal unidirectional covering sets. Technical Report arXiv:0901.3692v3 [cs.CC], ACM Computing Research Repository, CoRR (2009)Google Scholar
  20. 20.
    Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Raising NP lower bounds to parallel NP lower bounds. SIGACT News 28(2), 2–13 (1997)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. Journal of the ACM 44(6), 806–825 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dorothea Baumeister
    • 1
  • Felix Brandt
    • 2
  • Felix Fischer
    • 3
  • Jan Hoffmann
    • 2
  • Jörg Rothe
    • 1
  1. 1.Institut für InformatikUniversität DüsseldorfDüsseldorfGermany
  2. 2.Institut für InformatikUniversität MünchenMünchenGermany
  3. 3.Harvard School of Engineering and Applied SciencesCambridgeUSA

Personalised recommendations