On the computation of fully proportional representation

doi:10.1613/JAIR.3896

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On the computation of fully proportional representation

Journal Article•DOI•

On the computation of fully proportional representation

Nadja Betzler¹, Arkadii Slinko², Johannes Uhlmann¹•Institutions (2)

Technical University of Berlin¹, University of Auckland²

01 May 2013-Journal of Artificial Intelligence Research (AI Access Foundation)-Vol. 47, Iss: 1, pp 475-519

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.

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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.

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Book•DOI•

Handbook of Computational Social Choice

[...]

Felix Brandt¹, Vincent Conitzer², Ulle Endriss³, Jérôme Lang⁴, Jérôme Lang⁵, Ariel D. Procaccia⁶ - Show less +2 more•Institutions (6)

Technische Universität München¹, Durham University², University of Amsterdam³, Centre national de la recherche scientifique⁴, University of Paris⁵, Carnegie Mellon University⁶

25 Apr 2016-Research Papers in Economics

TL;DR: This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively and offers detailed introductions to each of the field's major themes.

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Abstract: The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.

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396 citations

Cites background or methods from "On the computation of fully proport..."

...The complexity class ΘP2 is the class of problems that can be solved, in polynomial time, by a (hypothetical) machine that has access to an oracle capable of deciding NP-complete problems in an instant, with the restriction that the number of queries to the oracle must be at most logarithmic in the size of the problem....
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...ΠP2 , a complexity class located at the second level of the polynomial hierarchy, is the class of decision problems for which a certificate for a negative answer can be verified in polynomial time by a machine that has access to an oracle for answering NP-complete problems in an instant....
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..., 2008) Winner determination for the Chamberlin-Courant scheme with the Borda misrepresentation function is NP-complete (Lu and Boutilier, 2011a) Winner determination for the minimax versions of the Chamberlin-Courant and Monroe schemes is NP-complete (Betzler et al., 2013)...
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...While we have seen that the premise-based rule can be manipulated, doing so is hard, at least in the worst case and for agendas involving large formulas:15 Proposition 11 (Endriss et al., 2012) The manipulation problem for the premise-based rule and judges with Hamming preferences is NP-complete....
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...Domain restrictions: for single-peaked profiles, most multiwinner problems discussed above become polynomial; the only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule (Betzler et al., 2013)....
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Book Chapter•

Multiwinner Voting: A New Challenge for Social Choice Theory

[...]

Piotr Faliszewski, Arkadii Slinko, Nimrod Talmon

30 Oct 2017

214 citations

Cites background or methods from "On the computation of fully proport..."

...On the other hand, Betzler et al. (2013) used the framework of parameterized complexity to show that winner determination for these rules can be solved efficiently for elections with few voters or with few alternatives....
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..., 2008), even when certain natural parameters of the election are small (Betzler et al., 2013) or when preferences of the voters are single-crossing (Skowron et al., 2015b) (hardness for single-peaked elections is known only for a more general variant of the rule (Betzler et al., 2013)). Yet, recently, Skowron et al. (2015a) proposed a greedy variant of this rule:...
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...…according to the Monroe rule is computationally hard (Procaccia et al., 2008), even when certain natural parameters of the election are small (Betzler et al., 2013) or when preferences of the voters are single-crossing (Skowron et al., 2015b) (hardness for single-peaked elections is known…...
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...On the other hand, Betzler et al. (2013) used the framework of parameterized complexity to show that winner determination for these rules can be solved efficiently for elections with few voters or with few alternatives. They also showed that these rules are polynomial-time computable for single-peaked elections, whereas Skowron et al. (2015b) have shown the same for single-crossing elections....
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..., 2008), even when certain natural parameters of the election are small (Betzler et al., 2013) or when preferences of the voters are single-crossing (Skowron et al....
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Posted Content•

Justified Representation in Approval-Based Committee Voting

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Haris Aziz¹, Markus Brill², Vincent Conitzer², Edith Elkind³, Rupert Freeman², Toby Walsh¹ - Show less +2 more•Institutions (3)

NICTA¹, Duke University², University of Oxford³

31 Jul 2014-arXiv: Multiagent Systems

TL;DR: In this article, a natural axiom for committee voting, called justified representation (JR), was proposed, which requires that if a large enough group of voters exhibits agreement by supporting the same candidate, then at least one voter in this group has an approved candidate in the winning committee.

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Abstract: We consider approval-based committee voting, i.e. the setting where each voter approves a subset of candidates, and these votes are then used to select a fixed-size set of winners (committee). We propose a natural axiom for this setting, which we call justified representation (JR). This axiom requires that if a large enough group of voters exhibits agreement by supporting the same candidate, then at least one voter in this group has an approved candidate in the winning committee. We show that for every list of ballots it is possible to select a committee that provides JR. However, it turns out that several prominent approval-based voting rules may fail to output such a committee. In particular, while Proportional Approval Voting (PAV) always outputs a committee that provides JR, Reweighted Approval Voting (RAV), a tractable approximation to PAV, does not have this property. We then introduce a stronger version of the JR axiom, which we call extended justified representation (EJR), and show that PAV satisfies EJR, while other rules we consider do not; indeed, EJR can be used to characterize PAV within the class of weighted PAV rules. We also consider several other questions related to JR and EJR, including the relationship between JR/EJR and core stability, and the complexity of the associated algorithmic problems.

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185 citations

Posted Content•

Properties of Multiwinner Voting Rules

[...]

Edith Elkind¹, Piotr Faliszewski, Piotr Skowron¹, Arkadii Slinko²•Institutions (2)

University of Oxford¹, University of Auckland²

09 Jun 2015-arXiv: Computer Science and Game Theory

TL;DR: This paper considers committee selection rules that can be viewed as generalizations of single-winner scoring rules, including SNTV, Bloc, k-Borda, STV, as well as several variants of the Chamberlin–Courant rule and the Monroe rule and their approximations.

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Abstract: The goal of this paper is to propose and study properties of multiwinner voting rules which can be consider as generalisations of single-winner scoring voting rules. We consider SNTV, Bloc, k-Borda, STV, and several variants of Chamberlin--Courant's and Monroe's rules and their approximations. We identify two broad natural classes of multiwinner score-based rules, and show that many of the existing rules can be captured by one or both of these approaches. We then formulate a number of desirable properties of multiwinner rules, and evaluate the rules we consider with respect to these properties.

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181 citations

Cites background or methods from "On the computation of fully proport..."

...Computational complexity of winner determination For -Monroe and -CC with ∈ { 1, min}, it is known that finding even a single winning committee is computationally hard (Procaccia et al. 2008; Lu and Boutilier 2011; Betzler et al. 2013)....
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...Unfortunately, these rules are hard to compute, irrespective of tie-breaking, both for the Borda satisfaction function (Lu and Boutilier 2011; Betzler et al. 2013) and for various approvalbased satisfaction functions (Procaccia et al. 2008; Betzler et al. 2013)....
[...]
...One way to classify them is according to the type of the input: there are preference-based rules (whose inputs are sequences of linear orders; see, e.g., the work of Brams and Fishburn (2002)), approval-based rules (whose inputs are sequences of dichotomies; see, e.g., the works of Brams et al.…...
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...The main idea is to provide an optimal assignment of committee members to voters, by using a satisfaction function to measure the quality of the assignment....
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...…be efficiently approximated, or admit polynomial-time algorithms when voters’ preferences are drawn from restricted domains (Lu and Boutilier 2011; Betzler et al. 2013; Cornaz et al. 2012; Yu et al. 2013; Skowron et al. 2015, 2016; Skowron and Faliszewski 2015; Skowron et al. 2015; Elkind and…...
[...]

Proceedings Article•

Finding a collective set of items: from proportional multirepresentation to group recommendation

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Piotr Skowron¹, Piotr Faliszewski, Jérôme Lang²•Institutions (2)

University of Warsaw¹, Paris Dauphine University²

25 Jan 2015

TL;DR: The problem is hard in general, but a number of tractability results for its natural special cases are shown.

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Abstract: We consider the following problem: There is a set of items (e.g., movies) and a group of agents (e.g., passengers on a plane); each agent has some intrinsic utility for each of the items. Our goal is to pick a set of K items that maximize the total derived utility of all the agents (i.e., in our example we are to pick K movies that we put on the plane's entertainment system). However, the actual utility that an agent derives from a given item is only a fraction of its intrinsic one, and this fraction depends on how the agent ranks the item among the chosen, available, ones. We provide a formal specification of the model and provide concrete examples and settings where it is applicable. We show that the problem is hard in general, but we show a number of tractability results for its natural special cases.

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171 citations

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References

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Computers and Intractability: A Guide to the Theory of NP-Completeness

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TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.

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Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.

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Abstract: From the Publisher: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects. In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage. As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds. Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.

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