19 research outputs found
PTAS for Minimax Approval Voting
We consider Approval Voting systems where each voter decides on a subset to
candidates he/she approves. We focus on the optimization problem of finding the
committee of fixed size k minimizing the maximal Hamming distance from a vote.
In this paper we give a PTAS for this problem and hence resolve the open
question raised by Carragianis et al. [AAAI'10]. The result is obtained by
adapting the techniques developed by Li et al. [JACM'02] originally used for
the less constrained Closest String problem. The technique relies on extracting
information and structural properties of constant size subsets of votes.Comment: 15 pages, 1 figur
Approximation and Parameterized Complexity of Minimax Approval Voting
We present three results on the complexity of Minimax Approval Voting. First,
we study Minimax Approval Voting parameterized by the Hamming distance from
the solution to the votes. We show Minimax Approval Voting admits no algorithm
running in time , unless the Exponential
Time Hypothesis (ETH) fails. This means that the
algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by
this, we then show a parameterized approximation scheme, running in time
, which is essentially
tight assuming ETH. Finally, we get a new polynomial-time randomized
approximation scheme for Minimax Approval Voting, which runs in time
,
almost matching the running time of the fastest known PTAS for Closest String
due to Ma and Sun [SIAM J. Comp. 2009].Comment: 14 pages, 3 figures, 2 pseudocode
Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation
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 items that
maximize the total derived utility of all the agents (i.e., in our example we
are to pick 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
Mathematical Programming formulations for the efficient solution of the -sum approval voting problem
In this paper we address the problem of electing a committee among a set of
candidates and on the basis of the preferences of a set of voters. We
consider the approval voting method in which each voter can approve as many
candidates as she/he likes by expressing a preference profile (boolean
-vector). In order to elect a committee, a voting rule must be established
to `transform' the voters' profiles into a winning committee. The problem
is widely studied in voting theory; for a variety of voting rules the problem
was shown to be computationally difficult and approximation algorithms and
heuristic techniques were proposed in the literature. In this paper we follow
an Ordered Weighted Averaging approach and study the -sum approval voting
(optimization) problem in the general case . For this problem we
provide different mathematical programming formulations that allow us to solve
it in an exact solution framework. We provide computational results showing
that our approach is efficient for medium-size test problems ( up to 200,
up to 60) since in all tested cases it was able to find the exact optimal
solution in very short computational times
Mathematical programming formulations for the efficient solution of the k-sum approval voting problem
In this paper we address the problem of electing a committee among a set of m candidates and on the basis of the preferences of a set of n voters. We consider the approval voting method in which each voter can approve as many candidates as she/he likes by expressing a preference profile (boolean m-vector). In order to elect a committee, a voting rule must be established to ‘transform’ the n voters’ profiles into a winning committee. The problem
is widely studied in voting theory; for a variety of voting rules the problem was shown to be computationally difficult and approximation algorithms and heuristic techniques were proposed in the literature. In this paper we follow an Ordered Weighted Averaging approach and study the k-sum approval voting (optimization) problem in the general case 1 ≤ k < n. For this problem we provide different mathematical programming formulations that allow us
to solve it in an exact solution framework. We provide computational results showing that our approach is efficient for medium-size test problems (n up to 200, m up to 60) since in all tested cases it was able to find the exact optimal solution in very short computational times.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona
Voting Rules for Expressing Conditional Preferences in Multiwinner Elections
Ο τομέας της Υπολογιστικής Θεωρίας Κοινωνικής Επιλογής μελετά, από αλγοριθμική σκοπιά, την αποτίμηση των προσωπικών προτιμήσεων προς μια συλλογική απόφαση. Πληθώρα προβλημάτων σε πολυπρακτορικά συστήματα, τεχνολογίες λήψης αποφάσεων, σχεδιασμό δικτύων, πολιτικό σχεδιασμό, συστήματα συστάσεων και άλλα, απαιτούν το σχεδιασμό και τη θεωρητική αξιολόγηση κανόνων ψηφοφορίας.
Στο πρώτο κεφάλαιο παρουσιάζουμε την προέλευση, ορισμένες εφαρμογές και υποπεριοχές μαζί με μία ιστορική επισκόπηση του αντικειμένου. Στο δεύτερο κεφάλαιο, εισάγουμε τον αναγνώστη σε εκλογικά σενάρια με περισσότερους από έναν νικητές, περιγράφοντας κάποιες επιθυμητές ιδιότητες των σχετικών κανόνων ψηφοφοριών και ορίζοντας τους πιο συχνά χρησιμοποιούμενους κανόνες μαζί με μία ματιά στα γνωστά αλγοριθμικά και υπολογιστικά τους αποτελέσματα. Μιας και σε πολλές περιπτώσεις, οι ψηφοφόροι επιθυμούν να τους επιτραπεί να εκφράσουν εξαρτήσεις μεταξύ των θεμάτων, όταν καλούνται να αποφασίσουν για περισσότερα από ένα θέματα, στο τρίτο κεφάλαιο εστιάζουμε σε εκλογές συνδυαστικής φύσεως, παρουσιάζοντας ορισμένες σχετικές εφαρμογές μαζί με λύσεις που έχουν προταθεί για την αντιμετώπιση αυτών των περιστάσεων. Τέλος, στο τέταρτο κεφάλαιο, περιγράφουμε ένα μοντέλο για χειρισμό ψήφων αποδοχής υπό συνθήκες σε πολλαπλά δυαδικά ζητήματα, ακολουθούμενο από ορισμένα νέα αποτελέσματα που αφορούν κυρίως βέλτιστους και προσεγγιστικούς αλγορίθμους για τον minisum και τον minimax κανόνα.Computational Social Choice studies the aggregation of individual preferences toward a collective decision from an algorithmic point of view. Various problems in multiagent systems, decision making technologies, network design, policy making, recommendation systems and so on, require the design and theoretical evaluation of a wide range of voting rules.
In the first chapter we present the origins, possible applications, some of the subtopics of Computational Social Choice as well as a historical overview of the field. In the second chapter we introduce the reader to election scenarios with more than a single winner by describing some commonly desired properties of multi-winner voting rules and defining the most widely used rules together with a glance at algorithmic and computational aspects. Since in many voting settings, voters wish to be allowed to express preferential dependencies, in the third chapter we focus on elections on combinatorial domains by presenting some specific applications along with some solutions which have been proposed in order to deal with combinatorial votes. Ultimately, in the fourth chapter we describe the recently proposed model for handling conditional approval preferences on multiple binary issues followed by new contributions which mainly concerns optimum and approximate results for minisum and minimax conditional approval voting rule
Multiwinner Voting with Fairness Constraints
Multiwinner voting rules are used to select a small representative subset of
candidates or items from a larger set given the preferences of voters. However,
if candidates have sensitive attributes such as gender or ethnicity (when
selecting a committee), or specified types such as political leaning (when
selecting a subset of news items), an algorithm that chooses a subset by
optimizing a multiwinner voting rule may be unbalanced in its selection -- it
may under or over represent a particular gender or political orientation in the
examples above. We introduce an algorithmic framework for multiwinner voting
problems when there is an additional requirement that the selected subset
should be "fair" with respect to a given set of attributes. Our framework
provides the flexibility to (1) specify fairness with respect to multiple,
non-disjoint attributes (e.g., ethnicity and gender) and (2) specify a score
function. We study the computational complexity of this constrained multiwinner
voting problem for monotone and submodular score functions and present several
approximation algorithms and matching hardness of approximation results for
various attribute group structure and types of score functions. We also present
simulations that suggest that adding fairness constraints may not affect the
scores significantly when compared to the unconstrained case.Comment: The conference version of this paper appears in IJCAI-ECAI 201
Parameterized Complexity of Multi-winner Determination: More Effort Towards Fixed-Parameter Tractability
We study the parameterized complexity of Winners Determination for three
prevalent -committee selection rules, namely the minimax approval voting
(MAV), the proportional approval voting (PAV), and the Chamberlin-Courant's
approval voting (CCAV). It is known that Winners Determination for these rules
is NP-hard. Moreover, these problems have been studied from the parameterized
complexity point of view with respect to some natural parameters recently.
However, many results turned out to be W[1]-hard or W[2]-hard. Aiming at
deriving more fixed-parameter algorithms, we revisit these problems by
considering more natural and important single parameters, combined parameters,
and structural parameters.Comment: 31 pages, 2 figures, AAMAS 201
On Computing Centroids According to the p-Norms of Hamming Distance Vectors
In this paper we consider the p-Norm Hamming Centroid problem which asks to determine whether some given strings have a centroid with a bound on the p-norm of its Hamming distances to the strings. Specifically, given a set S of strings and a real k, we consider the problem of determining whether there exists a string s^* with (sum_{s in S} d^{p}(s^*,s))^(1/p) <=k, where d(,) denotes the Hamming distance metric. This problem has important applications in data clustering and multi-winner committee elections, and is a generalization of the well-known polynomial-time solvable Consensus String (p=1) problem, as well as the NP-hard Closest String (p=infty) problem.
Our main result shows that the problem is NP-hard for all fixed rational p > 1, closing the gap for all rational values of p between 1 and infty. Under standard complexity assumptions the reduction also implies that the problem has no 2^o(n+m)-time or 2^o(k^(p/(p+1)))-time algorithm, where m denotes the number of input strings and n denotes the length of each string, for any fixed p > 1. The first bound matches a straightforward brute-force algorithm. The second bound is tight in the sense that for each fixed epsilon > 0, we provide a 2^(k^(p/((p+1))+epsilon))-time algorithm. In the last part of the paper, we complement our hardness result by presenting a fixed-parameter algorithm and a factor-2 approximation algorithm for the problem