Committee scoring voting rules are multiwinner analogues of positional scoring rules which constitute an important subclass of single-winner voting rules. We identify several natural subclasses of committee scoring rules, namely, weakly separable, representation-focused, top- @math -counting, OWA-based, and decomposable rules. We characterize SNTV, Bloc, and @math -Approval Chamberlin--Courant as the only nontrivial rules in pairwise intersections of these classes. We provide some axiomatic characterizations for these classes, where monotonicity properties appear to be especially useful. The class of decomposable rules is new to the literature. We show that it strictly contains the class of OWA-based rules and describe some of the applications of decomposable rules.
@cite_69: We investigate the existence of approximation algorithms for maximization of submodular functions, that run in a fixed parameter tractable (FPT) time. Given a non-decreasing submodular set function v:2 X →R v : 2 X → R the goal is to select a subset S of K elements from X such that v(S) v ( S ) is maximized. We identify three properties of set functions, referred to as p -separability properties, and we argue that many real-life problems can be expressed as maximization of submodular, p -separable functions, with low values of the parameter p . We present FPT approximation schemes for the minimization and maximization variants of the problem, for several parameters that depend on characteristics of the optimized set function, such as p and K .
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@cite_3: We establish a link between multiwinner elections and apportionment problems by showing how approval-based multiwinner election rules can be interpreted as methods of apportionment. We consider several multiwinner rules and observe that they induce apportionment methods that are well-established in the literature on proportional representation. For instance, we show that Proportional Approval Voting induces the D'Hondt method and that Monroe's rule induces the largest reminder method. We also consider properties of apportionment methods and exhibit multiwinner rules that induce apportionment methods satisfying these properties.
@cite_34: We study a generic framework that provides a unified view on two important classes of problems: (i) extensions of the k-median problem where clients are interested in having multiple facilities in their vicinity (e.g., due to the fact that, with some small probability, the closest facility might be malfunctioning and so might not be available for using), and (ii) finding winners according to some appealing multiwinner election rules, i.e., election system aimed for choosing representatives bodies, such as parliaments, based on preferences of a population of voters over individual candidates. Each problem in our framework is associated with a vector of weights: we show that the approximability of the problem depends on structural properties of these vectors. We specifically focus on the harmonic sequence of weights for which the objective function interpreted in a multiwinner election setup reflects to the well-known Proportional Approval Voting (PAV) rule. Our main result is that, due to the specific (harmonic) structure of weights, the problem allows constant factor approximation. This is surprising since the problem can be interpreted as a variant of the k-median problem where we do not assume that the connection costs satisfy the triangle inequality. The algorithm we propose is based on dependent rounding [Srinivasan, FOCS'01] applied to the solution of a natural LP-relaxation of the problem. The rounding process is well known to produce distributions over integral solutions satisfying Negative Correlation (NC), which is usually sufficient for the analysis of approximation guarantees offered by rounding procedures. In our analysis, however, we need to use the fact that the carefully implemented rounding process satisfies a stronger property, called Negative Association (NA), which allows us to apply standard concentration bounds for conditional random variables.
@cite_44: 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 @math items that maximize the total derived utility of all the agents (i.e., in our example we are to pick @math 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.
@cite_77: Many NP-hard winner determination problems admit polynomial-time algorithms when restricting inputs to be single-peaked. Commonly, such algorithms employ dynamic programming along the underlying axis. We introduce a new technique: carefully chosen integer linear programming (IP) formulations for certain voting problems admit an LP relaxation which is totally unimodular if preferences are single-peaked, and which thus admits an integral optimal solution. This technique gives fast algorithms for finding optimal committees under the PAV and Chamberlin-Courant voting rules under single-peaked preferences, as well as for certain OWA-based rules. Under single-crossing preferences, Young scores can also be calculated. An advantage of this technique is that no special-purpose algorithm needs to be used to exploit structure in the input preferences: any standard IP solver will terminate in the first iteration if the input is single-peaked, and will continue to work otherwise.
@cite_16: We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show that computing the winner for proportional approval voting is NP-hard, closing a long standing open problem. As none of the rules are strategyproof, even for dichotomous preferences, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents. In many settings, we show that it is NP-hard for an agent or agents to compute how best to vote given a fixed set of approval ballots from the other agents.
@cite_10: The issue of fair representation will take center stage as U.S. congressional districts are reapportioned based on the 2000 Census. Using U.S. history as a guide, the authors develop a theory of fair representation that establishes various principles for translating state populations --or vote totals of parties --into a fair allocation of congressional seats. They conclude that the current apportionment formula cheats the larger states in favor of the smaller, contrary to the intentions of the founding fathers and compromising the Supreme Court's ""one man, one vote"" rulings. Balinski and Young interweave the theoretical development with a rich historical account of controversies over representation, and show how many of these principles grew out of political contests in the course of United States history. The result is a work that is at once history, politics, and popular science. The book --updated with data from the 1980 and 1990 Census counts --vividly demonstrates that apportionment deals with the very substance of political power.
Proportional Approval Voting (PAV) received a bit less attention than the Chamberlin--Courant rule, but due to the work of @cite_10 on justified representation, it is now being studied with increasing interest (briefly put, @. have shown that PAV is remarkably good at providing committees that represent the voters proportionally, as also confirmed by @cite_3 ; see the work of @cite_4 for another rule with similar properties). The rule was shown to be @math -hard to compute @cite_16 @cite_44 , but the standard greedy @math -approximation algorithm works for it. Very recently, @cite_34 have shown a different, apparently much more powerful algorithm. The rule was also considered in the context of restricted domains @cite_77 . FPT approximation schemes for PAV and other OWA-based rules were provided by Skowron @cite_69 .