## Parameterized computational complexity of Dodgson and Young elections (2007)

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Citations: | 21 - 7 self |

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@TECHREPORT{Betzler07parameterizedcomputational,

author = {Nadja Betzler and Jiong Guo and Rolf Niedermeier},

title = {Parameterized computational complexity of Dodgson and Young elections},

institution = {},

year = {2007}

}

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### Abstract

Abstract. We show that, other than for standard complexity theory with known NP-completeness results, the computational complexity of the Dodgson and Young election systems is completely different from a parameterized complexity point of view. That is, on the one hand, we present an efficient fixed-parameter algorithm for determining a Condorcet winner in Dodgson elections by a minimum number of switches in the votes. On the other hand, we prove that the corresponding problem for Young elections, where one has to delete votes instead of performing switches, is W[2]-complete. In addition, we study Dodgson elections that allow ties between the candidates and give fixed-parameter tractability as well as W[2]-hardness results depending on the cost model for switching ties. 1

### Citations

871 | Parametrized Complexity
- Downey, Fellows
- 1999
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Citation Context ...practice”. This is the point of view of classical, “one-dimensional” computational complexity analysis. 2 By way of contrast, we propose the framework of parameterized computational complexity theory =-=[6,9, 16]-=- for studying election systems. 3 For Dodgson and Young elections, we consider the question whether the NPhard problems become fixed-parameter tractable with respect to the parameter number of editing... |

359 |
Parameterized Complexity Theory
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- 2006
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Citation Context ...practice”. This is the point of view of classical, “one-dimensional” computational complexity analysis. 2 By way of contrast, we propose the framework of parameterized computational complexity theory =-=[6,9, 16]-=- for studying election systems. 3 For Dodgson and Young elections, we consider the question whether the NPhard problems become fixed-parameter tractable with respect to the parameter number of editing... |

284 | Invitation to Fixed-Parameter Algorithms
- Niedermeier
- 2006
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Citation Context ...practice”. This is the point of view of classical, “one-dimensional” computational complexity analysis. 2 By way of contrast, we propose the framework of parameterized computational complexity theory =-=[6,9, 16]-=- for studying election systems. 3 For Dodgson and Young elections, we consider the question whether the NPhard problems become fixed-parameter tractable with respect to the parameter number of editing... |

234 |
Integer programming with a fixed number of variables
- Lenstra
- 1983
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Citation Context ...o give information about the (exponential terms of the) corresponding running times. Herein, “ILP” means that fixed-parameter tractability follows by an integer linear program and a result of Lenstra =-=[19]-=- implying impractical running times. Bold-faced results are new, the FPT-results for the parameter “number of candidates” can be directly obtained from [1,26], and the FPT-results for the Young electi... |

142 |
Voting Schemes for Which It Can Be Difficult to Tell Who Won the Election. Social Choice and Welfare 6(2): 157–165
- Bartholdi, Tovey, et al.
- 1989
(Show Context)
Citation Context ...ner. In Young elections, the editing operation is to remove a vote, again trying to minimize the number of removals in order to end up with a Condorcet winner. In their seminal work, Bartholdi et al. =-=[1]-=- initiated the study of the computational complexity of election systems. They showed that to decide whether a distinguished candidate can be made a Condorcet winner by performing no more than a given... |

120 |
When Are Elections with Few Candidates Hard to Manipulate
- Conitzer, Sandholm, et al.
- 2007
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Citation Context ...nsted [13]. corresponding winner and ranking problems are even complete for Θ p 2 1 Also known as the writer Lewis Carroll. 2 For more classical complexity results w.r.t. election systems we refer to =-=[5, 10]-=-. 3 In companion work [2], we also do so for Kemeny elections. 4 In parameterized algorithmics [6, 9, 16] the solution size typically is the “standard parameter”. 5 In the meantime, Fellows et al. ind... |

88 |
Essai sur l'application de l'analyse `a la probabilit'e des decisions rendues `a la pluralite des voix
- Condorcet
- 1785
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Citation Context ...assic election system consists of a set of candidates and a set of vote(r)s. Each voter chooses an order of preference (total order) among the candidates. The well-known Condorcet principle from 1785 =-=[6]-=- then requires that a winner of an election is the candidate who is preferred to each other candidate in more than half of the votes. Unfortunately, a Condorcet winner does not always exist. Hence, se... |

83 |
Invitation to data reduction and problem kernelization
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Citation Context ...questions directly arising from our work. Regarding Dodgson Score, it would be interesting to investigate the power of polynomial-time data reduction and the existence of (small) problem kernels (see =-=[14]-=- for a survey on problem kernels which play a key role in parameterized algorithmics). Moreover, there are open questions regarding other parameterizations (see also Table 1): Bartholdi et al. [1] gav... |

54 | Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP
- Hemaspaandra, Hemaspaandra, et al.
- 1997
(Show Context)
Citation Context ...dorcet winner by performing no more than a given number of editing operations is NP-complete for both Dodgson and Young elections. In a further breakthrough, for Dodgson elections Hemaspaandra et al. =-=[11]-=- and later for Young elections Rothe et al. [18] showed that the , the class of problems that can be solved via parallel access to NP. Thus, Faliszewski et al. [7] concluded that “since checking wheth... |

51 | Anyone but him: The complexity of precluding an alternative
- Hemaspaandra, Hemaspaandra, et al.
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Citation Context ...step towards finding an optimal 2 Also known as the writer Lewis Carroll. 3 In the meantime, there are many publications presenting classical complexity results for election systems, for example, see =-=[7,15,17,23,27]-=-. 2Table 1 Parameterized complexity of Dodgson Score and (Dual) Young Score with respect to different parameters. In case of fixed-parameter tractability we also give information about the (exponenti... |

50 |
Dichotomy for voting systems
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Citation Context ...nsted [13]. corresponding winner and ranking problems are even complete for Θ p 2 1 Also known as the writer Lewis Carroll. 2 For more classical complexity results w.r.t. election systems we refer to =-=[5, 10]-=-. 3 In companion work [2], we also do so for Kemeny elections. 4 In parameterized algorithmics [6, 9, 16] the solution size typically is the “standard parameter”. 5 In the meantime, Fellows et al. ind... |

46 | Algorithms for the coalitional manipulation problem
- Zuckerman, Procaccia, et al.
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Citation Context ...step towards finding an optimal 2 Also known as the writer Lewis Carroll. 3 In the meantime, there are many publications presenting classical complexity results for election systems, for example, see =-=[7,15,17,23,27]-=-. 2Table 1 Parameterized complexity of Dodgson Score and (Dual) Young Score with respect to different parameters. In case of fixed-parameter tractability we also give information about the (exponenti... |

44 | A short introduction to Computational Social Choice
- Chevaleyre, Endriss, et al.
- 2007
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Citation Context ...hing ties. 1 Introduction Computational social choice and, more specifically, the computational complexity of election systems has become an increasingly important field of interdisciplinary research =-=[3, 7]-=-. Besides the obvious application in computational politics where the fact that people have varying preferences concerning whom to elect leads to preference aggregation demands via some election syste... |

34 |
Exact complexity of the winner problem for Young elections
- Rothe, Spakowski, et al.
(Show Context)
Citation Context ... number of editing operations is NP-complete for both Dodgson and Young elections. In a further breakthrough, for Dodgson elections Hemaspaandra et al. [11] and later for Young elections Rothe et al. =-=[18]-=- showed that the , the class of problems that can be solved via parallel access to NP. Thus, Faliszewski et al. [7] concluded that “since checking whether a given candidate has won should be in polyno... |

29 | A richer understanding of the complexity of election systems - Faliszewski, Hemaspaandra, et al. - 2009 |

22 | On the Approximability of Dodgson and Young Elections
- Caragiannis, Covey, et al.
(Show Context)
Citation Context ...g elections. This stands in sharp contrast to traditional complexity analysis, where both election systems appear as equally hard [1,16,25], and complements results on polynomial-time approximability =-=[3]-=-. Furthermore, we found that the complexities of Dodgson elections allowing ties between the candidates strongly vary (fixed-parameter tractability vs W[2]completeness) depending on the cost model for... |

18 |
Extending Condorcet’s rule
- Young
- 1977
(Show Context)
Citation Context ...l with certain “editing problems”. In this work, we focus on two classic editing problems from social choice theory [22], one due to C. L. Dodgson 2 from 1876 [8] and one due to H. P. Young from 1977 =-=[26]-=-. In Dodgson elections, the editing operation is to switch neighboring candidates in the voters’ preference lists and the goal is to minimize the overall number of switches needed in order to result i... |

16 | Fixed-parameter algorithms for Kemeny scores - Betzler, Fellows, et al. |

16 |
A method of taking votes on more than two issues. Pamphlet printed by the Clarendon Press, Oxford, and headed “not yet published” (see the discussions in [MU95,Bla58], both of which reprint this paper
- Dodgson
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Citation Context ...n other words, these election systems deal with certain “editing problems”. In this work, we focus on two classic editing problems from social choice theory [22], one due to C. L. Dodgson 2 from 1876 =-=[8]-=- and one due to H. P. Young from 1977 [26]. In Dodgson elections, the editing operation is to switch neighboring candidates in the voters’ preference lists and the goal is to minimize the overall numb... |

15 | Guarantees for the Success Frequency of an Algorithm for Finding Dodgson Election Winners
- Homan, Hemaspaandra
- 2006
(Show Context)
Citation Context ...tractable with a worse running time [8]. 2Proc. 11th SWAT, 2008 Moreover, this complements recent work on a simple greedy heuristic for finding Dodgson winners with a guaranteed frequency of success =-=[12]-=- and some work on the approximability of Dodgson and Young elections [14,17]. In particular, Procaccia et al. [17] gave (randomized) approximation algorithms for Dodgson elections and show that it is ... |

13 |
On complexity of lobbying in multiple referenda
- Christian, Fellows, et al.
- 2007
(Show Context)
Citation Context ...W[2]-complete. Our results imply that Dodgson elections can be put in actual use whenever the input instances are close to having a Condorcet winner. This answers an open question of Christian et al. =-=[4]-=- 5 and refutes a parameterized hardness conjecture of McCabe-Dansted [13]. corresponding winner and ranking problems are even complete for Θ p 2 1 Also known as the writer Lewis Carroll. 2 For more cl... |

13 | Complexity of Strategic Behavior in Multi-Winner Elections
- Meir, Procaccia, et al.
- 2008
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Citation Context ...step towards finding an optimal 2 Also known as the writer Lewis Carroll. 3 In the meantime, there are many publications presenting classical complexity results for election systems, for example, see =-=[7,15,17,23,27]-=-. 2Table 1 Parameterized complexity of Dodgson Score and (Dual) Young Score with respect to different parameters. In case of fixed-parameter tractability we also give information about the (exponenti... |

11 | F.A.: Fixedparameter algorithms for Kemeny rankings
- Betzler, Fellows, et al.
- 2009
(Show Context)
Citation Context ...2 remain NP-complete (which easily follows from the NP-completeness of the case without ties [1]), their parameterized complexity differs. The problem DTS2 is fixed-parameter tractable while DTS1 is W=-=[2]-=-complete. To show the fixed-parameter tractability of DTS2 one can use a slight modification of algorithm DodScore from Figure 1. However, for DTS2 we obtain a slightly worse running time than for Dod... |

7 |
Approximability and Computational Feasibility of Dodgson’s Rule. Master’s thesis. The University of Auckland Complexity of Lobbying in Multiple Referenda 9
- McCabe-Dansted
- 2006
(Show Context)
Citation Context ...tual use whenever the input instances are close to having a Condorcet winner. This answers an open question of Christian et al. [4] 5 and refutes a parameterized hardness conjecture of McCabe-Dansted =-=[13]-=-. corresponding winner and ranking problems are even complete for Θ p 2 1 Also known as the writer Lewis Carroll. 2 For more classical complexity results w.r.t. election systems we refer to [5, 10]. 3... |

4 | Approximability and inapproximability of Dodgson and Young elections
- Procaccia, Feldman, et al.
- 2007
(Show Context)
Citation Context ..., this complements recent work on a simple greedy heuristic for finding Dodgson winners with a guaranteed frequency of success [12] and some work on the approximability of Dodgson and Young elections =-=[14,17]-=-. In particular, Procaccia et al. [17] gave (randomized) approximation algorithms for Dodgson elections and show that it is hard to approximate Young elections by any factor. Moreover, for Dodgson ele... |

2 |
Approximability of Dodgson’s rule. Social Choice and Welfare, 2007. To appear; available electronically at SpringerLink; a preliminary appers in [MPS06
- McCabe-Dansted, Pritchard, et al.
(Show Context)
Citation Context ..., this complements recent work on a simple greedy heuristic for finding Dodgson winners with a guaranteed frequency of success [12] and some work on the approximability of Dodgson and Young elections =-=[14,17]-=-. In particular, Procaccia et al. [17] gave (randomized) approximation algorithms for Dodgson elections and show that it is hard to approximate Young elections by any factor. Moreover, for Dodgson ele... |

2 |
Threshold dominating sets and an improved version of W[2
- Downey, Fellows
- 1998
(Show Context)
Citation Context ...r of switches as the parameter. More precisely, we show containment in W[2] by giving a parameterized reduction to a variant of the k-Weighted Circuit Satisfiability (k-WCS) problem that defines W[2] =-=[9]-=-. The k-WCS problem has as input a circuit and a positive integer k, and asks whether it has a weight-k satisfying assignment (an assignment setting the values of exactly k input gates to 1). Here, we... |