Unweighted coalitional manipulation under the Borda rule Is NP-hard

IJCAI, pp. 55-60, 2011.

Cited by: 89|Bibtex|Views130|Links
EI
Keywords:
unweighted coalitional manipulationborda rulecertain numberopen problemborda winnerMore(6+)
Weibo:
We showed NP-hardness for BORDA MANIPULATION even for very restricted settings such as having constant numbers of input votes and manipulators

Abstract:

The Borda voting rule is a positional scoring rule where, for m candidates, for every vote the first candidate receives m- 1 points, the second m- 2 points and so on. A Borda winner is a candidate with highest total score. It has been a prominent open problem to determine the computational complexity of UNWEIGHTED COALITIONAL MANIPULATION...More

Code:

Data:

Introduction
  • In their recent overview on “AI’s war on manipulation” Faliszewski and Procaccia [2010] write “An enigmatic open problem is the complexity of Unweighted Coalitional Manipulation under Borda.” Here, we settle this open problem by showing NP-hardness for UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA,1 which we subsequently refer to as BORDA MANIPULATION.
  • Zuckerman et al [2009] showed that for BORDA MANIPULATION a greedy algorithm can always find a set of x manipulators if the given input allows x − 1 manipulators to make a distinguished candidate win.
  • In other words, this means that the optimization version of BORDA MANIPULATION is polynomial-time approximable with an additive error one.
  • WEIGHTED COALITIONAL MANIPULATION UNDER BORDA is known to be NP-hard even for three candidates [Conitzer et al, 2007; Hemaspaandra and Hemaspaandra, 2007]
Highlights
  • In their recent overview on “AI’s war on manipulation” Faliszewski and Procaccia [2010] write “An enigmatic open problem is the complexity of Unweighted Coalitional Manipulation under Borda.” Here, we settle this open problem by showing NP-hardness for UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA,1 which we subsequently refer to as BORDA MANIPULATION
  • We show that BORDA MANIPULATION remains NP-hard for three input votes and for any other number greater than three
  • We investigate the structure of instances resulting from the NP-hardness reduction
  • We showed NP-hardness for BORDA MANIPULATION even for very restricted settings such as having constant numbers of input votes and manipulators
  • Our NP-hardness proof is of theoretical nature in the sense that it is a purely worst-case result with little impact on practical aspects of solving BORDA MANIPULATION
Conclusion
  • The authors showed NP-hardness for BORDA MANIPULATION even for very restricted settings such as having constant numbers of input votes and manipulators.
  • The authors' NP-hardness proof is of theoretical nature in the sense that it is a purely worst-case result with little impact on practical aspects of solving BORDA MANIPULATION.
  • This motivates the issue of parameterizing NP-hard problems such as BORDA MANIPULATION in the spirit of multivariate algorithmics [Niedermeier, 2010].
  • It is of interest whether in case of two manipulators one can solve the problem in less than O(|C|!) time
Summary
  • Introduction:

    In their recent overview on “AI’s war on manipulation” Faliszewski and Procaccia [2010] write “An enigmatic open problem is the complexity of Unweighted Coalitional Manipulation under Borda.” Here, we settle this open problem by showing NP-hardness for UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA,1 which we subsequently refer to as BORDA MANIPULATION.
  • Zuckerman et al [2009] showed that for BORDA MANIPULATION a greedy algorithm can always find a set of x manipulators if the given input allows x − 1 manipulators to make a distinguished candidate win.
  • In other words, this means that the optimization version of BORDA MANIPULATION is polynomial-time approximable with an additive error one.
  • WEIGHTED COALITIONAL MANIPULATION UNDER BORDA is known to be NP-hard even for three candidates [Conitzer et al, 2007; Hemaspaandra and Hemaspaandra, 2007]
  • Conclusion:

    The authors showed NP-hardness for BORDA MANIPULATION even for very restricted settings such as having constant numbers of input votes and manipulators.
  • The authors' NP-hardness proof is of theoretical nature in the sense that it is a purely worst-case result with little impact on practical aspects of solving BORDA MANIPULATION.
  • This motivates the issue of parameterizing NP-hard problems such as BORDA MANIPULATION in the spirit of multivariate algorithmics [Niedermeier, 2010].
  • It is of interest whether in case of two manipulators one can solve the problem in less than O(|C|!) time
Tables
  • Table1: Two manipulative votes v1 and v2 illustrating the strategy used in the proof of Proposition 4.2
Download tables as Excel
Funding
  • NB was supported by the DFG project “PAWS”, NI 369/10
Reference
  • [Bartholdi III et al., 1989] J. J. Bartholdi III, C. A. Tovey, and M. A. Trick. The computational difficulty of manipulating an election. Social Choice and Welfare, 6:227–241, 1989.
    Google ScholarLocate open access versionFindings
  • [Betzler et al., 2009] N. Betzler, S. Hemmann, and R. Niedermeier. A multivariate complexity analysis of determining possible winners given incomplete votes. In Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI), pages 53–58, 2009.
    Google ScholarLocate open access versionFindings
  • [Conitzer et al., 2007] V. Conitzer, T. Sandholm, and J. Lang. When are elections with few candidates hard to manipulate? Journal of the ACM, 54(3):1–33, 2007.
    Google ScholarLocate open access versionFindings
  • [Davies et al., 2010] J. Davies, G. Katsirelos, N. Narodytska, and T. Walsh. An empirical study of Borda manipulation. In Proceedings of the 3rd International Workshop on Computational Social Choice, pages 91–102, 2010.
    Google ScholarLocate open access versionFindings
  • [Davies et al., 2011] J. Davies, G. Katsirelos, N. Narodytska, and T. Walsh. Complexity of and algorithms for Borda manipulation. Under review for AAAI 2011, 2011.
    Google ScholarFindings
  • [Dorn and Schlotter, 2010] B. Dorn and I. Schlotter. Multivariate complexity analysis of swap bribery. In Proceedings of the 5th International Symposium on Parameterized and Exact Computation (IPEC), volume 6478 of LNCS, pages 107–122.
    Google ScholarLocate open access versionFindings
  • [Faliszewski and Procaccia, 2010] P. Faliszewski and A. Procaccia. AI’s war on manipulation: Are we winning? AI Magazine, 31(4):53–64, 2010.
    Google ScholarLocate open access versionFindings
  • [Faliszewski et al., 2010] P. Faliszewski, E. Hemaspaandra, and L. A. Hemaspaandra. Using complexity to protect elections. Communications of the ACM, 53(1):74–82, 2010.
    Google ScholarLocate open access versionFindings
  • [Hemaspaandra and Hemaspaandra, 2007] E. Hemaspaandra and L. A. Hemaspaandra. Dichotomy for voting systems. Journal of Computer and System Sciences, 73(1):73–83, 2007.
    Google ScholarLocate open access versionFindings
  • [Lenstra, 1983] H. W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538–548, 1983.
    Google ScholarLocate open access versionFindings
  • [Niedermeier, 2010] R. Niedermeier. Reflections on multivariate algorithmics and problem parameterization. In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS’10), volume 5 of LIPIcs, pages 17–32, 2010.
    Google ScholarLocate open access versionFindings
  • [Procaccia and Rosenschein, 2007] A. Procaccia and J. S. Rosenschein. Junta distributions and the average-case complexity of manipulating elections. Journal of Artificial Intelligence Research, 28:157–181, 2007.
    Google ScholarLocate open access versionFindings
  • [Walsh, 2010] T. Walsh. Is computational complexity a barrier to manipulation? In Proceedings of the 11th International Workshop on Computational Logic in MultiAgent Systems (CLIMA), volume 6245 of LNCS, pages 1– 7.
    Google ScholarLocate open access versionFindings
  • [Xia and Conitzer, 2008] L. Xia and V. Conitzer. Generalized scoring rules and the frequency of coalitional manipulability. In Proceedings of the 9th ACM Conference on Electronic Commerce (EC), pages 109–118, 2008.
    Google ScholarLocate open access versionFindings
  • [Xia et al., 2010] L. Xia, V. Conitzer, and A. D. Procaccia. A scheduling approach to coalitional manipulation. In Proceedings of the 11th ACM Conference on Electronic Commerce (EC), pages 275–284. ACM, 2010.
    Google ScholarLocate open access versionFindings
  • [Yu et al., 2004] W. Yu, H. Hoogeveen, and J. K. Lenstra. Minimizing makespan in a two-machine flow shop with delays and unit-time operations is NP-hard. Journal of Scheduling, 7:333–348, 2004.
    Google ScholarLocate open access versionFindings
  • [Yu, 1996] W. Yu. Personal communication. 1996.
    Google ScholarFindings
  • [Zuckerman et al., 2009] M. Zuckerman, A. D. Procaccia, and J. S. Rosenschein. Algorithms for the coalitional manipulation problem. Artificial Intelligence, 173(2):392– 412, 2009.
    Google ScholarLocate open access versionFindings
Your rating :
0

 

Best Paper
Best Paper of IJCAI, 2011
Tags
Comments