Proceedings Article•
Finding a collective set of items: from proportional multirepresentation to group recommendation
25 Jan 2015-pp 2131-2137
TL;DR: The problem is hard in general, but a number of tractability results for its natural special cases are shown.
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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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.
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.
396 citations
Book Chapter•
30 Oct 2017
214 citations
Cites background or methods from "Finding a collective set of items: ..."
...Proportional Approval Voting (PAV)....
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...In each round it adds to S a candidate c with the maximal value of ∑ i : c∈Ai 1 |S∩Ai|+1 , i.e., a candidate c which maximizes the PAV score of S ∪ {c}....
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...For an overview of the internal structure of such rules we point the reader to the works of Faliszewski et al. (2016a,b); axiomatic characterization of these rules is due to Skowron et al. (2016b)....
[...]
...Brill et al. (2017), on the other hand, discussed a relation between multiwinner voting rules and methods of apportionment, which allows to view PAV and RAV as extensions of the d’Hondt method of apportionment to the multiwinner setting (see Chapter 3 of this book for more details on seat allocations)....
[...]
...Yet, RAV can be viewed as a Multiwinner Voting: A New Challenge for Social Choice Theory 43 good approximation algorithm for PAV (Skowron et al., 2016a) which can be even better approximated when certain natural parameters are low (Skowron, 2016)....
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Posted Content•
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.
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.
185 citations
Posted Content•
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.
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.
181 citations
25 Jan 2015
TL;DR: A natural axiom is proposed for 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), which is called justified representation ($$\mathrm {JR}$$JR).
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. We then check if this axiom is fulfilled by well-known approval-based voting rules. We show that the answer is negative for most of the rules we consider, with notable exceptions of PAV (Proportional Approval Voting), an extreme version of RAV (Reweighted Approval Voting), and, for a restricted preference domain, MAV (Minimax Approval Voting). 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 do not. We also consider several other questions related to JR and EJR, including the relationship between JR/EJR and unanimity, and the complexity of the associated algorithmic problems.
148 citations
References
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01 Jan 1979
42,654 citations
Book•
01 Jan 1979
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.
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.
40,020 citations
03 Jan 1988
TL;DR: A type of operator for aggregation called an ordered weighted aggregation (OWA) operator is introduced and its performance is found to be between those obtained using the AND operator and the OR operator.
Abstract: The author is primarily concerned with the problem of aggregating multicriteria to form an overall decision function. He introduces a type of operator for aggregation called an ordered weighted aggregation (OWA) operator and investigates the properties of this operator. The OWA's performance is found to be between those obtained using the AND operator, which requires all criteria to be satisfied, and the OR operator, which requires at least one criteria to be satisfied. >
6,534 citations
"Finding a collective set of items: ..." refers background in this paper
...This distinction between the intrinsic value of an item and its value distorted by its rank are also considered in several other research fields, especially decision theory (“rank-dependent utility theory”) and multicriteria decision making, from which we borrow one of the main ingredients of our approach, ordered weighted average (OWA) operators (Yager 1988) (see the following formal model; the reader might want to consult the work of Kacprzyk et al....
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...To this end, (1) for each agent i ∈ N and for each item aj ∈ A, we have an intrinsic utility ui,aj , ui,aj ≥ 0, that agent i derives from aj ; (2) the utility that each agent derives from a set of K items is an ordered weighted average (Yager 1988) of this agent’s intrinsic utilities for these items....
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TL;DR: It is shown that a “greedy” heuristic always produces a solution whose value is at least 1 −[(K − 1/K]K times the optimal value, which can be achieved for eachK and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm.
Abstract: LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)źz(SźT)+z(SźT) for allS, T inN. Such a function is called submodular. We consider the problem maxSźN{a(S):|S|≤K,z(S) submodular}.
Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.
We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a "greedy" heuristic always produces a solution whose value is at least 1 ź[(K ź 1)/K]K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e ź 1)/e, where e is the base of the natural logarithm.
4,103 citations
"Finding a collective set of items: ..." refers methods in this paper
...The next result follows by applying the result of Nemhauser et al. (1978) to function uut and Algorithm 1....
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...This follows by applying the famous result of Nemhauser et al. (1978) for nondecreasing submodular set functions to the case of uut (recall Definition 1)....
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Posted Content•
01 Jan 1978
TL;DR: In this article, the authors considered the problem of finding a maximum weight independent set in a matroid, where the elements of the matroid are colored and the items of the independent set can have no more than K colors.
Abstract: LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)gez(SxcupT)+z(SxcapT) for allS, T inN. Such a function is called submodular. We consider the problem maxSsubN{a(S):|S|leK,z(S) submodular}. Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem. We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a ldquogreedyrdquo heuristic always produces a solution whose value is at least 1 –[(K – 1)/K] K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e – 1)/e, where e is the base of the natural logarithm.
3,351 citations