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

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.

Justified representation in approval-based committee voting

“…3 Generalizations of PAV with w 1 = 0 have been considered by Fishburn and Pekec (2004) and Skowron et al (2015d). We note that such rules do not satisfy justified representation (as defined in Section 3).…”

“…For some of these rules, namely, AV, SAV, RAV, GAV and GAV T , a winning committee can be computed in polynomial time; this is also true for w-RAV as long the entries of the weight vector are rational numbers that can be efficiently computed given the number of candidates. In contrast, PAV, MAV, and MonAV are computationally hard (Aziz et al 2015;Skowron et al 2015a;LeGrand et al 2007;; for w-PAV, the hardness result holds for most weight vectors, including (1, 0, . .…”

“…Misra et al (2015) show that MAV is fixed-parameter tractable for a number of natural parameters; Elkind and Lackner (2015) obtain fixed parameter tractability results for PAV when voters' preferences are, in some sense, single-dimensional. There is also a number of tractability results for CCAV, and, to a lesser extent, for MonAV; we refer the reader to the work of Skowron et al (2015a) and references therein.…”

“…However, both PAV and MAV admit efficient approximation algorithms (i.e., algorithms that output committees which are approximately optimal with respect to the optimization criteria of these rules) and have been analyzed from the perspective of parameterized complexity. Specifically, w-PAV admits an efficient 1 − 1 e -approximation algorithm as long as the weight vector w is efficiently computable and non-increasing; in fact, such an algorithm is provided by w-RAV (Skowron et al 2015a). For MAV, LeGrand et al (2007) propose a simple 3-approximation algorithm; Caragiannis et al (2010) improve the approximation ratio to 2 and Byrka and Sornat (2014) develop a polynomial-time approximation scheme.…”

“…While the most-studied scenario is that of selecting a single candidate out of many, it is often the case that one needs to select a fixed-size set of winners (committee): this includes domains such as parliamentary elections, the hiring of faculty members, or (automated) agents deciding on a set of plans (LeGrand et al 2007;Davis et al 2014;Elkind et al 2014Skowron et al 2015a). The study of algorithmic complexity of voting rules that output committees is an active research direction Meir et al 2008;Caragiannis et al 2010;Lu and Boutilier 2011;Cornaz et al 2012;Betzler et al 2013;Skowron et al 2015b,c).…”