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Proceedings ArticleDOI

Committee selection with multimodal preferences

TL;DR: This work designs efficient algorithms for certain cases of committee selection with multimodal preferences and discusses applications of the model and the computational complexity of several generalizations of known committee scoring rules to this setting.
Abstract: We study committee selection with multimodal preferences: Assuming a set of candidatesA, a set of voters V , and ` layers, where each voter v ∈ V has ordinal preferences over the alternatives for each layer separately, the task is to select a committee S ⊆ A of size k. We discuss applications of our model and study the computational complexity of several generalizations of known committee scoring rules (specifically, k-Borda and Chamberlin–Courant) to our setting, as well as discuss domain restrictions for our model. While most problems we encounter are computationally intractable in general, we nevertheless design efficient algorithms for certain cases.
Citations
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Proceedings ArticleDOI
TL;DR: In this article , the authors introduce three position-based matching models, which minimize the "dissatisfaction score", which measures matchings from different perspectives, and present diverse complexity results for these three models, among others, polynomial-time solvability for the first model.
Abstract: Many models have been proposed for computing a one-to-one matching between two equal-sized sets/sides of agents, each assigned with one preference list of the agents in the opposite side. The most prominent one might be the Stable Matching model. Re-cently, the Stable Matching model has been extended to the multimodal setting [6, 13, 29], where each agent has more than one preference lists, each representing a criterion based on which the agents of the opposite side are evaluated. We use a layer to denote the set of preference lists of agents, which are based on the same criterion. Thus, the single modal matching problem has only one layer. This setting finds applications in many real-world scenarios. However, it turns out that stable matchings might not exist with multi-modal preferences and the determination is NP-hard and W-hard with respect to several natural parameters. Here, we introduce three position-based matching models, which minimize the “dissatisfaction score”. We define four dissatisfaction scores, which measure matchings from different perspectives. The first model minimizes the total respective dissatisfaction score over all layers, while the second minimizes the maximum of the respective score over all layers. The third model seeks for a matching 𝑀 which is Layer Pareto-optimal, meaning that there does not exist a matching 𝑀 ′ , which is at least as good as 𝑀 with respect to the respective dissatisfaction score in all layers, but is strictly better in at least one layer. We present diverse complexity results for these three models, among others, polynomial-time solvability for the first model. We also investigate the generalization which given an upper bound on the dissatisfaction score, computes a matching involving subsets of agents and a subset of layers. Hereby, we mainly focus on the parameterized complexity with respect to parameters such as the size of agent subsets, or the size of the layer subset and achieve fixed-parameter tractability as well as intractability results.

2 citations

Proceedings ArticleDOI
16 May 2022
TL;DR: In this paper , the complexity of stable matching problems with multilayer approval preferences was studied and eleven stability notions derived from three well-established stability notions for stable matchings with ties and four adaptions proposed by Chen et al.
Abstract: We study stable matching problems where agents have multilayer preferences: There are ℓ layers each consisting of one preference relation for each agent. Recently, Chen et al. [EC ’18] studied such problems with strict preferences, establishing four multilayer adaptions of classical notions of stability. We follow up on their work by analyzing the computational complexity of stable matching problems with multilayer approval preferences. We consider eleven stability notions derived from three well-established stability notions for stable matchings with ties and the four adaptions proposed by Chen et al. For each stability notion, we show that the problem of finding a stable matching is either polynomial-time solvable or NP-hard. Furthermore, we examine the influence of the number of layers and the desired “degree of stability” on the problems’ complexity. Somewhat surprisingly, we discover that assuming approval preferences instead of strict preferences does not consider-ably simplify the situation (and sometimes even makes polynomial-time solvable problems NP-hard).
Proceedings ArticleDOI
TL;DR: In this article , the distortion in attribute approval committee elections is studied, where each candidate satisfies a variety of attributes in different categories (e.g., academic degree, work experience, lo-cation).
Abstract: In attribute approval elections, the task is to select sets of winning candidates, while each candidate satisfies a variety of attributes in different categories (e.g., academic degree, work experience, lo-cation). Every voter specifies, which attributes in each category are desirable for a candidate, whereas each candidate might satisfy only some of the attributes. In this paper, we study questions of distortion in attribute approval committee elections. We introduce different methods to derive approval ballots, ordinal preferences, or cardinal preferences from a given attribute approval ballot. Then for a given voting method, assuming only a derived preference is provided, we compute the ratio of the voters’ satisfaction for the worst possible committee, with the satisfaction of the actual winning committee, given the attribute approval ballots.
Journal ArticleDOI
TL;DR: In this paper , the authors considered a new model of election control that by assigning different rules to the votes from different layers, makes the special candidate p being the winner of the election (a rule can be assigned to different layers).
Abstract: We study the election control problem with multi-votes, where each voter can present a single vote according different views (or layers, we use"layer"to represent"view"). For example, according to the attributes of candidates, such as: education, hobby or the relationship of candidates, a voter may present different preferences for the same candidate set. Here, we consider a new model of election control that by assigning different rules to the votes from different layers, makes the special candidate p being the winner of the election (a rule can be assigned to different layers). Assuming a set of candidates C among a special candidate"p", a set of voters V, and t layers, each voter gives t votes over all candidates, one for each layer, a set of voting rules R, the task is to find an assignment of rules to each layer that p is acceptable for voters (possible winner of the election). Three models are considered (denoted as sum-model, max-model, and min-model) to measure the satisfaction of each voter. In this paper, we analyze the computational complexity of finding such a rule assignment, including classical complexity and parameterized complexity. It is interesting to find out that 1) it is NP-hard even if there are only two voters in the sum-model, or there are only two rules in sum-model and max-model; 2) it is intractable with the number of layers as parameter for all of three models; 3) even the satisfaction of each vote is set as dichotomous, 1 or 0, it remains hard to find out an acceptable rule assignment. Furthermore, we also get some other intractable and tractable results.
References
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Journal ArticleDOI
TL;DR: In this article, the authors provide a type of reasoning which will contribute to the development of the theory of tradeunions, the firm, and the cartel; and provide the basis for a theory of the equilibrium distribution of taxation or of public expenditure.
Abstract: When a decision is reached by voting or is arrived at by a group all of whose members are not in complete accord, there is no part of economic theory which applies. This paper is intended to help fill this gap; to provide a type of reasoning which will contribute to the development of the theory of tradeunions, the firm, and the cartel; and to provide the basis for a theory of the equilibrium distribution of taxation or of public expenditure. Still other uses of the theory might be not less important. For reasons of space we avoid discussion of many points that demand fuller treatment and only attempt to indicate the course of the argument.

2,195 citations


"Committee selection with multimodal..." refers background in this paper

  • ...A particularly popular domain restriction is the single-peaked domain, originally proposed by Black [6]....

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Journal ArticleDOI
TL;DR: This work shows that INDEPENDENT SET is complete for W, and the W Hierarchy of parameterized problems was defined, and complete problems were identified for the classes W [ t ] for t ⩾ 2.

659 citations


"Committee selection with multimodal..." refers methods in this paper

  • ...We provide a polynomial time reduction from the W[1]hard problem Independent Set (IS) [12]....

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  • ...Towards this, we give a reduction from the W[1]-hard problem Independent Set (IS) [12], in which given a graph G and an integer t; we shall decide the existence of a t-sized set X ⊆ V (G) containing only nonadjacent vertices....

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Journal ArticleDOI
TL;DR: It is demonstrated that winner selection in two prominent proportional representation voting systems is a computationally intractable problem—implying that these systems are impractical when the assembly is large, and in settings where the size of the Assembly is constant, the problem can be solved in polynomial time.
Abstract: We demonstrate that winner selection in two prominent proportional representation voting systems is a computationally intractable problem—implying that these systems are impractical when the assembly is large. On a different note, in settings where the size of the assembly is constant, we show that the problem can be solved in polynomial time.

220 citations


"Committee selection with multimodal..." refers background in this paper

  • ...Due to Proposition 4 and the fact that Egalitarian-CC is NP-hard and W[2]-hard w.r.t. the committee size k [5], we have following result....

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  • ...In particular, while finding winning committees under kBorda can be done in polynomial time (one has to select k candidates with the highest individual Borda scores), CC is NP-hard [26] but FPTfor certain parameters, admit approximation algorithms, and certain heuristics are known to be effective for it [5, 20, 28, 15]....

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  • ...Due to Proposition 3 and the fact that CC is NP-hard and W[2]-hard w.r.t. k [5], we obtain the following result....

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  • ...Corollary 8 Min-CC, Sum-CC, and Vector-CC are NP-hard and W[2]-hard w.r.t. k even for Local-SP, and n = 1 We next study the computational complexity of Max-CC for Global-SP profiles, and obtain intractability....

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  • ...Due to Proposition 2, and NP-hardness and W[2]-hardness of EgalitarianCC with respect to the committee size k [5], we obtain: Corollary 4 Min-CC is NP-hard and W[2]-hard w.r.t. k even for n = 1....

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Journal ArticleDOI
TL;DR: This work investigates two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe and investigates the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters.
Abstract: We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

215 citations


"Committee selection with multimodal..." refers background or methods in this paper

  • ...We adapt the corresponding algorithm for CC [5]: We guess a clustering of the voters (i....

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  • ...Due to Proposition 2, and NP-hardness and W[2]-hardness of EgalitarianCC with respect to the committee size k [5], we obtain:...

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  • ...[5] for Max-CC and Min-CC under Global-SP; unfortunately,...

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  • ...In particular, a polynomial time algorithm, computing winning committees under CC for single-peaked profiles is known [5]....

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  • ...k [5], we obtain the following result....

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Book Chapter
30 Oct 2017

214 citations


"Committee selection with multimodal..." refers background in this paper

  • ..., [16] and [17]), k-Borda and CC represent two extreme multiwinner rules....

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  • ...As k-Borda is in P [17], we have the following:...

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  • ...We obtain tractability in polynomial time; due to Proposition 1 and the fact that k-Borda can be solved in polynomial time [17], we have the following result....

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