### Citations

903 |
Combinatorial Optimization: Polyhedra and Efficiency,
- Schrijver
- 2003
(Show Context)
Citation Context ...}), (2) because we use respectively: Lemma 1 and the fact that Z has at least one element; definition of ina(·) for empty set and upperbound for ina(·) function; assumption that Y = ∅. 1 according to =-=[11]-=-, f : 2S 7→ R is supermodular iff ∀Y,Z⊆S f(Y ) + f(Z) 6 f(Y ∪ Z) + f(Y ∩ Z) which is equivalent with ∀Y⊆Z⊆S ∀s∈S f(Z)− f(Z ∪ {s}) 6 f(Y )− f(Y ∪ {s}). Case 3: Z 6= ∅, Y 6= ∅: From definition of ina(·)... |

293 |
Probabilistic construction of deterministic algorithms: Approximate packing integer programs.
- Raghavan
- 1988
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Citation Context ...mate solution with probability at least 13 . We may derandomize the algorithm analogously to how it was done in the PTAS for the Closest String problem [8]. For more on derandomization techniques see =-=[10]-=-. 5 Algorithm and its complexity analysis Now we are ready to combine the ideas into a single algorithm. Algorithm ALG(R) Input: S = {s1, s2, . . . , sn} ∈ ({0, 1}m)n, 0 6 k 6 m,R ∈ N>1 Output: sALG... |

65 | On the closest string and substring problems.
- Li, Ma, et al.
- 2002
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Citation Context ... obtained approximation ratio essentially matches the integrality gap of the LP. In this paper we give a PTAS for the Minimax Approval Voting problem. Our work is based on the PTAS for Closest String =-=[8]-=-, which is a similar problem to MAV but there we do not have the restriction on the number of 1’s in the result. Technically, our contribution is the method of handling the number of 1’s in the output... |

11 | Some results on approximating the minimax solution in approval voting.
- LeGrand, Markakis, et al.
- 2007
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Citation Context ...an easy task and can be done by simply selecting the k candidates approved by the largest number of voters. By contrast, Minimax Approval Voting was shown by LeGrand [6] to be NP-hard. LeGrand et al. =-=[7]-=- obtained 3-approximation by a very simple k-completion algorithm. Next, Carragianis et al. [5] gave the currently best 2-approximation algorithm. The algorithm was obtained by rounding a fractional s... |

9 | Computational aspects of multi-winner approval voting
- Aziz, Gaspers, et al.
- 2014
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Citation Context ...lated work and our results Many different objective functions have been proposed and studied in the context of selecting the committee based on the set of votes collected in an Approval Voting system =-=[1,2]-=-. Clearly, optimizing the sum of Hamming distances to all votes ? jby@cs.uni.wroc.pl ?? krzysztof.sornat@cs.uni.wroc.pl ar X iv :1 40 7. 72 16 v2s[ cs .D S]s2 9 S eps20 14 is an easy task and can be d... |

9 | A minimax procedure for electing committees.
- Brams, Kilgour, et al.
- 2007
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Citation Context ...lt is obtained by applying a predefined election rule to the set of collected votes. In this paper we study the problem of implementing an appropriate election rule and focus on the Minimax objective =-=[3]-=-: we minimize the biggest dissatisfaction over voters. The resulting optimization problem is denoted MAV , and it is to select a committee composed of exactly k candidates, and minimizing the maximal ... |

8 | Approximation algorithms and mechanism design for minimax approval voting.
- Caragiannis, Kalaitzis, et al.
- 2010
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Citation Context ...r of voters. By contrast, Minimax Approval Voting was shown by LeGrand [6] to be NP-hard. LeGrand et al. [7] obtained 3-approximation by a very simple k-completion algorithm. Next, Carragianis et al. =-=[5]-=- gave the currently best 2-approximation algorithm. The algorithm was obtained by rounding a fractional solution to the natural LP relaxation of the problem, and obtained approximation ratio essential... |

6 |
Analysis of the minimax procedure
- LeGrand
- 2004
(Show Context)
Citation Context ...6 v2s[ cs .D S]s2 9 S eps20 14 is an easy task and can be done by simply selecting the k candidates approved by the largest number of voters. By contrast, Minimax Approval Voting was shown by LeGrand =-=[6]-=- to be NP-hard. LeGrand et al. [7] obtained 3-approximation by a very simple k-completion algorithm. Next, Carragianis et al. [5] gave the currently best 2-approximation algorithm. The algorithm was o... |

3 |
Approval Voting. 2nd edition
- Brams, Fishburn
- 2007
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Citation Context ...rained Closest String problem. The technique relies on extracting information and structural properties of constant size subsets of votes. 1 Introduction Approval Voting systems are widely considered =-=[2]-=- as an alternative to traditional elections, where each voter may select and support at most some small number of candidates. In Approval Voting each voter decides about every single candidate if he a... |

1 |
PTAS for Minimax Approval Voting. arXiv preprint arXiv:1407.7216v2
- Byrka, Sornat
- 2014
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Citation Context ...r ∈ {2, 3, . . . , n} we take Sr = Sr−1 ∪ {sir} where sir is such a vote that after adding it the inaccuracy function decreases the most, i.e., sir = arg max s∈S\Sr−1 ( ina(Sr−1)− ina(Sr−1 ∪ {s}) ) . =-=(4)-=- r0 r R m OPT ina(Sr) 6 OPTR 6 OPTR 6 OPTR 6 OPTR Fig. 1. The ina(·) function for the sequence of subsets S1 ⊂ S2 ⊂ . . . ⊂ Sn = S. We have min r∈{1,2,...,R} ina(Sr)− ina(Sr+1) 6 1 R ( R∑ r=1 ina(Sr)−... |

1 |
Randomized Algoritms
- Motvani, Raghavan
- 1995
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Citation Context ...[j] + χ(s′i[j] = 1) · ( 1− (s′)LP [j])) def.= def. = d ( (s′)LP , s′i ) (15) 6 qLP − d(s′′ALG, s′′i ). (20) d(s′, s′i) is a sum of β independent 0-1 variables. For ′ ∈ (0, 1) using Chernoff’s bound =-=[9]-=- we have: P ( d(s′, s′i) > (1 + ′) · E [ d(s′, s′i) ]) 6 exp ( −1 3 (′)2 · E[d(s′, s′i)]) . If we take ′ = 2·q IP E[d(s′,s′i)] then we obtain: exp ( −1 3 · (2) 2 · (qIP )2 E [ d(s′, s′i) ] ) > P(... |