## Approximability and Inapproximability of Dodgson and Young Elections (2007)

Citations: | 5 - 1 self |

### BibTeX

@MISC{Procaccia07approximabilityand,

author = {Ariel D. Procaccia and Michal Feldman and Jeffrey S. Rosenschein},

title = {Approximability and Inapproximability of Dodgson and Young Elections},

year = {2007}

}

### OpenURL

### Abstract

The voting rules proposed by Dodgson and Young are both designed to find the candidate closest to being a Condorcet winner, according to two different notions of proximity; the score of a given candidate is known to be hard to compute under both rules. In this paper, we put forward an LP-based randomized rounding algorithm which yields an O(log m) approximation ratio for the Dodgson score, where m is the number of candidates. Surprisingly, we show that the seemingly simpler Young score is N P-hard to approximate by any factor.

### Citations

11492 |
Computers and intractability, a guide to the theory of np-completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...s is given by a nonempty set of voters for which c ∗ is a Condorcet winner. In order to show N P-hardness, we present a polynomial-time reduction from the N P-hard Exact Cover by 3-Sets (X3C) problem =-=[8]-=- to our problem. An instance of the X3C problem includes a finite set of elements U, |U| = n (where n is divisible by 3), and a collection C of 3-element subsets of U, C = {C1, . . . , Ck}, such that ... |

359 |
The Theory of Committees and Elections
- Black
- 1958
(Show Context)
Citation Context ...is fairly easy to see that the preferences of the majority may be cyclic, hence a Condorcet winner does not necessarily exist. This unfortunate phenomenon is known as the Condorcet paradox (see Black =-=[4]-=-). In order to circumvent this result, several researchers have proposed to choose a candidate which is “as close as possible” to a Condorcet winner. Different notions of proximity can be chosen, and ... |

247 | Integer programming with a fixed number of variables
- Lenstra
- 1983
(Show Context)
Citation Context ...nd variables in their program depends purely on the number of candidates. Therefore, if the number of candidates is constant, the program is solvable in polynomial time using the algorithm of Lenstra =-=[14]-=-. However, if the number of candidates is not constant, the LP is of gargantuan size. 2 Fortunately, it is straightforward to modify the abovementioned ILP to obtain a program of polynomial size. As b... |

167 | Aggregating inconsistent information: Ranking and clustering
- Ailon, Charikar, et al.
- 2005
(Show Context)
Citation Context ... approximation of the Dodgson score, as the score of a Condorcet winner is 0, and is indeed the case here). Related work. The agenda of approximating voting rules was recently pursued by Ailon et al. =-=[1]-=-, Coppersmith et al. [7], and Kenyon-Mathieu and Schudy [12]. These works deal, directly or indirectly, with the Kemeny rank aggregation rule, which chooses a ranking of the candidates instead of a si... |

157 | Linear programming: Foundations and extensions
- Vanderbei
- 1997
(Show Context)
Citation Context ... 1 ∀a ∈ A \ {c ∗ }, � j i,j ∀i, j, xij ∈ {0, 1} xije a ij ≥ d(a) This ILP can be relaxed by asking merely that 0 ≤ xij ≤ 1 for all i and j. The resulting linear program (LP) can be solved efficiently =-=[19]-=-. We are now ready to present our randomized rounding algorithm. Its input and output are as before. Randomized Rounding Algorithm: 1. Solve the relaxed LP given by (1) to obtain a solution �x. 2. For... |

148 |
Voting schemes for which it can be difficult to tell who won the election
- Bartholdi, Tovey, et al.
- 1989
(Show Context)
Citation Context ...er by removing the least number of voters. Though these two voting rules sound appealing and straightforward, they are notoriously complicated to resolve. As early as 1989, Bartholdi, Tovey and Trick =-=[2]-=- have shown that computing the Dodgson score ∗ School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel, email: arielpro@cs.huji.ac.il. † School of Busin... |

60 |
Ordering by weighted number of wins gives a good ranking for weighted tournaments
- Coppersmith, Fleischer, et al.
- 2006
(Show Context)
Citation Context ...dgson score, as the score of a Condorcet winner is 0, and is indeed the case here). Related work. The agenda of approximating voting rules was recently pursued by Ailon et al. [1], Coppersmith et al. =-=[7]-=-, and Kenyon-Mathieu and Schudy [12]. These works deal, directly or indirectly, with the Kemeny rank aggregation rule, which chooses a ranking of the candidates instead of a single winning candidate. ... |

60 | 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 ... P-complete, and that pinpointing a Dodgson winner is N P-hard. This important paper was one of the first to introduce complexity-theoretic considerations to social choice theory. Hemaspaandra et al. =-=[9]-=- refined the abovementioned result by showing that the Dodgson winner problem is complete for Θ p 2 , the class of problems which can be solved by O(log n) queries to an N P set. Consequently, Rothe e... |

57 | How to rank with few errors
- Kenyon-Mathieu, Schudy
- 2007
(Show Context)
Citation Context ...orcet winner is 0, and is indeed the case here). Related work. The agenda of approximating voting rules was recently pursued by Ailon et al. [1], Coppersmith et al. [7], and Kenyon-Mathieu and Schudy =-=[12]-=-. These works deal, directly or indirectly, with the Kemeny rank aggregation rule, which chooses a ranking of the candidates instead of a single winning candidate. The Kemeny rule picks the ranking wh... |

48 | Improved bounds for computing Kemeny rankings
- Conitzer, Davenport, et al.
- 2006
(Show Context)
Citation Context ...n. More distantly related to our work is research which is concerned with exactly resolving hard-to-compute voting rules by heuristic methods. Typical examples include works regarding the Kemeny rule =-=[6]-=- and the Slater rule [5]. Another more remotely related field of research is concerned with finding approximate, efficient representations of voting rules, by eliciting as little information as possib... |

44 | Computing Slater rankings using similarities among candidates
- Conitzer
- 2006
(Show Context)
Citation Context ...d to our work is research which is concerned with exactly resolving hard-to-compute voting rules by heuristic methods. Typical examples include works regarding the Kemeny rule [6] and the Slater rule =-=[5]-=-. Another more remotely related field of research is concerned with finding approximate, efficient representations of voting rules, by eliciting as little information as possible; this line of researc... |

37 |
Monotone convergence of binomial probabilities and a generalization of Ramanujan’s
- JOGDEQ, SAMUELS
- 1968
(Show Context)
Citation Context ... i pi. Moreover, by the constraint ∀a ∈ A \ {c∗ }, � i,j xijea � ij ≥ d(a), we have that i pi ≥ d(a). We now employ a deceivingly intuitive but highly nontrivial result: Lemma 3.4 (Jogdeo and Samuels =-=[11]-=-). Let Y1, . . . , Yn be independent heterogeneous Bernoulli trials. Suppose that E[ � i Yi] is an integer. Then � � �� � � Pr Yi < E < 1/2. i Since d(a) is an integer, and E[ � i Yi] = � i pi ≥ d(a),... |

36 | Exact complexity of the winner problem for Young elections
- Rothe, Spakowski, et al.
- 2003
(Show Context)
Citation Context ...ed the abovementioned result by showing that the Dodgson winner problem is complete for Θ p 2 , the class of problems which can be solved by O(log n) queries to an N P set. Consequently, Rothe et al. =-=[18]-=- proved that the Young winner problem is also complete for Θ p 2 . The abovementioned complexity results give rise to the agenda of approximately calculating a candidate’s score, under the Dodgson and... |

23 | Parameterized Computational Complexity of Dodgson and Young Elections - Betzler, Guo, et al. - 2007 |

17 | Guarantees for the success frequency of an algorithm for finding Dodgsonelection winners
- Homan, Hemaspaandra
(Show Context)
Citation Context ...Dodgson ranking. This implies that approximation algorithms for Kemeny cannot be leveraged to approximate Dodgson. Two recent works have directly put forward algorithms for the Dodgson winner problem =-=[10, 15]-=-. Both papers independently build upon the same basic idea: if the number of voters is significantly larger than the number of candidates, and one looks at a uniform distribution over the preferences ... |

17 |
Extending Condorcet’s rule
- Young
- 1977
(Show Context)
Citation Context ...s between adjacent candidates in the voters’ rankings one has to introduce in order to make the given candidate a Condorcet winner. A Dodgson winner is any candidate with minimal Dodgson score. Young =-=[20]-=- raised a second option: measuring the distance by voters. Specifically, the Young score of a candidate is the size of the largest subset of voters such that, if only these ballots are taken into acco... |

14 | Learning voting trees
- Procaccia, Zohar, et al.
- 2007
(Show Context)
Citation Context ...esearch is concerned with finding approximate, efficient representations of voting rules, by eliciting as little information as possible; this line of research employs techniques from learning theory =-=[17, 16]-=-. Our results. Our results are two-fold. In the context of approximating the Dodgson score, we devise an O(log m) randomized approximation algorithm, where m is the number of candidates. Our algorithm... |

13 | Feasibility and Approximability of Dodgson’s rule
- McCabe-Dansted
- 2006
(Show Context)
Citation Context ...Dodgson ranking. This implies that approximation algorithms for Kemeny cannot be leveraged to approximate Dodgson. Two recent works have directly put forward algorithms for the Dodgson winner problem =-=[10, 15]-=-. Both papers independently build upon the same basic idea: if the number of voters is significantly larger than the number of candidates, and one looks at a uniform distribution over the preferences ... |

8 |
The Dodgson ranking and its relation to Kemenys method and Slaters rule, Social Choice and Welfare 23(1):91–102
- Klamler
- 2004
(Show Context)
Citation Context ...tion; Kenyon-Mathieu and Schudy further improve the approximation, and obtain a PTAS. Coppersmith et al. show that the Borda ranking is a 5-approximation of the Kemeny ranking. Interestingly, Klamler =-=[13]-=- discusses the relation between the Kemeny rule and an extension of Dodgson’s rule. However, Klamler shows that the candidate ranked first by Kemeny can appear anywhere in the Dodgson ranking. This im... |

4 | Automated design of voting rules by learning from examples
- Procaccia, Zohar, et al.
- 2006
(Show Context)
Citation Context ...esearch is concerned with finding approximate, efficient representations of voting rules, by eliciting as little information as possible; this line of research employs techniques from learning theory =-=[17, 16]-=-. Our results. Our results are two-fold. In the context of approximating the Dodgson score, we devise an O(log m) randomized approximation algorithm, where m is the number of candidates. Our algorithm... |