Parameterized Algorithms for Finding a Collective Set of Items

Abstract

We extend the work of Skowron et al. (AIJ, 2016) by considering the parameterized complexity of the following problem. We are given a set of items and a set of agents, where each agent assigns an integer utility value to each item. The goal is to find a set of k items that these agents would collectively use. For each such collective set of items, each agent provides a score that can be described using an OWA (ordered weighted average) operator and we seek a set with the highest total score. We focus on the parameterization by the number of agents and we find numerous fixed-parameter tractability results (however, we also find some W[1]-hardness results). It turns out that most of our algorithms even apply to the setting where each agent has an integer weight.

More Effort Towards Multiagent Knapsack

“…Since Committee Selection problem is W[1]-hard with respect to n for λ > 1 [10], we have the following result.…”

“…Due to Bredereck et al [10], Best-UK is FPT(n, λ) when the item costs are equal to 1. We extend their algorithm to solve both Median-UK and Best-UK.…”

“…For a start, we show that both the problems are NP-hard for every λ (Theorem 1). Since the Committee Selection problem with median scoring function is W[1]-hard with respect to n, the number of voters, for all λ > 1 [10], it follows that its generalization Median-UK must be as well (Corollary 1). Hence, even though an FPT 4 algorithm with respect to n is unlikely for Median-UK, we are able to present an XP algorithm with respect to n (Theorem 2), that runs in time O((m(λ+1)) n poly(n, m)).…”

Line-Up Elections: Parallel Voting with Shared Candidate Pool

“…Among this class of rules, we focus on the following four natural ones, quite well studied in other contexts, such as finding a collective set of items [5,9,12,26]:…”

Using Multiwinner Voting to Search for Movies

“…Based on this local election, for each user and each movie that he or she likes, we determine a utility score which indicates how relevant the movie is (briefly put, we need to distinguish between movies that are globally very popular, such as, e.g., The Lord of the Rings, from the ones that are mostly popular among the agents in the local election). Finally, we seek a winning committee with respect to one of the OWA-based multiwinner voting rules discussed by Skowron et al (2016), and output its contents as our result (see also the works of Aziz et al (2017) and Bredereck et al (2020) for further discussions of these rules). Since, in general, our rules are NP-hard to compute, we use approximation algorithms and heuristics.…”