Mathematical Programming formulations for the efficient solution of the kk-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 $1 \leq k <n$. 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

Towards cooperative urban traffic management: Investigating voting for travel groups

In den letzten Jahrzehnten haben intelligente Verkehrssysteme an Bedeutung gewonnen. Wir betrachten einen Teilbereich des kooperativen Verkehrsmanagements, nämlich kollektive Entscheidungsfindung in Gruppen von Verkehrsteilnehmern. In dem uns interessierenden Szenario werden Touristen, die eine Stadt besuchen, gebeten, Reisegruppen zu bilden und sich auf gemeinsame Besuchsziele (Points of Interest) zu einigen. Wir konzentrieren uns auf Wählen als Gruppenentscheidungsverfahren. Unsere Fragestellung ist, wie sich verschiedene Algorithmen zur Bildung von Reisegruppen und zur Bestimmung gemeinsamer Reiseziele hinsichtlich der System- und Benutzerziele unterscheiden, wobei wir als Systemziel große Gruppen und als Benutzerziele hohe präferenzbasierte Zufriedenheit und geringen organisatorischen Aufwand definieren. Wir streben an, einen Kompromiss zwischen System- und Benutzerzielen zu erreichen. Neu ist, dass wir die inhärenten Auswirkungen verschiedener Wahlregeln, Wahlprotokolle und Gruppenbildungsalgorithmen auf Benutzer- und Systemziele untersuchen. Altere Arbeiten zur kollektiven Entscheidungsfindung im Verkehr konzentrieren sich auf andere Zielgrößen, betrachten nicht die Gruppenbildung, vergleichen nicht die Auswirkungen mehrerer Wahlalgorithmen, benutzen andere Wahlalgorithmen, berücksichtigen nicht klar definierte Gruppen von Verkehrsteilnehmern, verwenden Wahlen für andere Anwendungen oder betrachten andere Algorithmen zur kollektiven Entscheidungsfindung als Wahlen. Wir untersuchen in der Hauptsimulationsreihe verschiedene Gruppenbildungsalgorithmen, Wahlprotokolle und Komiteewahlregeln. Wir betrachten sequentielle Gruppenbildung vs. koordinierte Gruppenbildung, Basisprotokoll vs. iteratives Protokoll und die Komiteewahlregeln Minisum-Approval, Minimax-Approval und Minisum-Ranksum. Die Simulationen wurden mit dem neu entwickelten Simulationswerkzeug LightVoting durchgef¨uhrt, das auf dem Multi-Agenten-Framework LightJason basiert. Die Experimente der Hauptsimulationsreihe zeigen, dass die Komiteewahlregel Minisum-Ranksum in den meisten Fällen bessere oder ebenso gute Ergebnisse erzielt wie die Komiteewahlregeln Minisum-Approval und Minimax-Approval. Das iterative Protokoll tendiert dazu, eine Verbesserung hinsichtlich der präferenzbasierten Zufriedenheit zu erbringen, auf Kosten einer deutlichen Verschlechterung hinsichtlich der Gruppengröße. Die koordinierte Gruppenbildung tendiert dazu, eine Verbesserung hinsichtlich der präferenzbasierten Zufriedenheit zu erbringen bei relativ geringen Kosten in Bezug auf die Gruppengröße. Dies führt uns dazu, die Komiteewahlregel Minisum-Ranksum, das Basisprotokoll und die koordinierte Gruppenbildung zu empfehlen, um einen Kompromiss zwischen System- und Benutzerzielen zu erreichen. Wir demonstrieren auch die Auswirkungen verschiedener Kombinationen von Gruppenbildungsalgorithmen und Wahlprotokollen auf die Reisekosten. Hier bietet die Kombination aus Basisprotokoll und koordinierter Gruppenbildung einen Kompromiss zwischen der präferenzbasierten Zufriedenheit und den Reisekosten. Zusätzlich zur Hauptsimulationsreihe bieten wir ein erweitertes Modell an, das die Präferenzen der Reisenden generiert, indem es die Attraktivität der möglichen Ziele und Distanzkosten, basierend auf den Entfernungen zwischen den möglichen Zielen, kombiniert. Als weiteren Anwendungsfall von Wahlverfahren betrachten wir ein Verfahren zur Treffpunktempfehlung, bei dem eine Bewertungs-Wahlregel und eine Minimax-Wahlregel zur Bestimmung von Treffpunkten verwendet werden. Bei kleineren Gruppen ist die durchschnittliche maximale Reisezeit unter der Bewertungs-Wahlregel deutlich höher. Bei größeren Gruppen nimmt der Unterschied ab. Bei kleineren Gruppen ist die durchschnittliche Verspätung für die Gruppe unter der Minimax-Wahlregel hoch, bei größeren Gruppen nimmt sie ab. Es ist also sinnvoll für kleinere Gruppen, die Minimax-Wahlregel zu verwenden, wenn man eine fairere Verteilung der Reisezeiten anstrebt, und die Bewertungs-Wahlregel zu verwenden, wenn das Ziel stattdessen ist, Verzögerungen für die Gruppe zu vermeiden. Für zukünftige Arbeiten wäre es sinnvoll, das Simulationskonzept anzupassen, um reale Bedingungen und Anforderungen berücksichtigen zu können. Weitere Möglichkeiten für zukünftige Arbeiten wären die Betrachtung zusätzlicher Algorithmen und Modelle, wie zum Beispiel die Betrachtung kombinatorischer Wahlen oder die Durchführung von Simulationen auf der Grundlage des erweiterten Modells, die Berücksichtigung der Rolle finanzieller Anreize zur Förderung von Ridesharing oder Platooning und die Nutzung des LightVoting-Tools für weitere Forschungsanwendungen.In the last decades, intelligent transport systems have gained importance. We consider a subarea of cooperative traffic management, namely collective decision-making in groups of traffic participants. In the scenario we are studying, tourists visiting a city are asked to form travel groups and to agree on common points of interest. We focus on voting as a collective decision-making process. Our question is how different algorithms for the formation of travel groups and for determining common travel destinations differ with respect to system and user goals, where we define as system goal large groups and as user goals high preference satisfaction and low organisational effort. We aim at achieving a compromise between system and user goals. What is new is that we investigate the inherent effects of different voting rules, voting protocols and grouping algorithms on user and system goals. Older works on collective decision-making in traffic focus on other target quantities, do not consider group formation, do not compare the effects of several voting algorithms, use other voting algorithms, do not consider clearly defined groups of vehicles, use voting for other applications or use other collective decision-making algorithms than voting. In the main simulation series, we examine different grouping algorithms, voting protocols and committee voting rules. We consider sequential grouping vs. coordinated grouping, basic protocol vs. iterative protocol and the committee voting rules Minisum-Approval, Minimax-Approval and Minisum-Ranksum. The simulations were conducted using the newly developed simulation tool LightVoting, which is based on the multi-agent framework LightJason. The experiments of the main simulation series show that the committee voting rule Minisum-Ranksum in most cases yields better than or as good results as the committee voting rules Minisum-Approval and Minimax-Approval. The iterative protocol tends to yield an improvement regarding preference satisfaction, at the cost of strong deterioriation regarding the group size. The coordinated grouping tends to yield an improvement regarding the preference satisfaction at relative small cost regarding the group size. This leads us to recommend the committee voting rule Minisum-Ranksum, the basic protocol and coordinated grouping in order to achieve a compromise between system and user goals. We also demonstrate the effect of different combinations of grouping algorithms and voting protocols on travel costs. Here, the combination of the basic protocol and coordinated grouping yields a compromise between preference satisfaction and traveller costs. Additionally to the main simulation series, we provide an extended model which generates traveller preferences by combining attractiveness of the points of interest and distance costs based on the distances between the points of interest. As further application of voting, we consider a meeting-point scenario where a range voting rule and a minimax voting rule are used to agree on meeting points. For smaller groups, the average maximum travel time is clearly higher for range voting. For larger groups, the difference decreases. For smaller groups, the average lateness for the group using minimax voting is high, for larger groups it decreases. Hence, it makes sense for smaller groups to use the minimax voting rule if one aims at fairer distribution of travel times, and to use the range voting rule if the goal is instead to avoid delay for the group. For future work, it would be useful to adapt the simulation concept to take real-world conditions and requirements into account. Further possibilities for future work would be considering additional algorithms and models, such as considering combinatorial voting or running simulations based on the extended model, considering the role of financial incentives to encourage ridesharing or platooning and using the LightVoting tool for further research applications

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

On the Complexity of Winner Determination and Strategic Control in Conditional Approval Voting

We focus on a generalization of the classic Minisum approval voting rule, introduced by Barrot and Lang (2016), and referred to as Conditional Minisum (CMS), for multi-issue elections with preferential dependencies. Under this rule, voters are allowed to declare dependencies between different issues, but the price we have to pay for this higher level of expressiveness is that we end up with a computationally hard rule. Motivated by this, we first focus on finding special cases that admit efficient algorithms for CMS. Our main result in this direction is that we identify the condition of bounded treewidth (of an appropriate graph, emerging from the provided ballots) as the necessary and sufficient condition for exact polynomial algorithms, under common complexity assumptions. We then move to the design of approximation algorithms. For the (still hard) case of binary issues, we identify natural restrictions on the voters' ballots, under which we provide the first multiplicative approximation algorithms for the problem. The restrictions involve upper bounds on the number of dependencies an issue can have on the others and on the number of alternatives per issue that a voter can approve. Finally, we also investigate the complexity of problems related to the strategic control of conditional approval elections by adding or deleting either voters or alternatives and we show that in most variants of these problems, CMS is computationally resistant against control. Overall, we conclude that CMS can be viewed as a solution that achieves a satisfactory tradeoff between expressiveness and computational efficiency, when we have a limited number of dependencies among issues, while at the same time exhibiting sufficient resistance to control

Multi-Winner Voting with Approval Preferences

Approval-based committee (ABC) rules are voting rules that output a fixed-size subset of candidates, a so-called committee. ABC rules select committees based on dichotomous preferences, i.e., a voter either approves or disapproves a candidate. This simple type of preferences makes ABC rules widely suitable for practical use. In this book, we summarize the current understanding of ABC rules from the viewpoint of computational social choice. The main focus is on axiomatic analysis, algorithmic results, and relevant applications.Comment: This is a draft of the upcoming book "Multi-Winner Voting with Approval Preferences

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

A review of network location theory and models

Cataloged from PDF version of article.In this study, we review the existing literature on network location problems. The study has a broad scope that includes problems featuring desirable and undesirable facilities, point facilities and extensive facilities, monopolistic and competitive markets, and single or multiple objectives. Deterministic and stochastic models as well as robust models are covered. Demand data aggregation is also discussed. More than 500 papers in this area are reviewed and critical issues, research directions, and problem extensions are emphasized.Erdoğan, Damla SelinM.S