Fair Knapsack

TL;DR

This paper explores a multiagent version of the knapsack problem, focusing on preference aggregation methods to select socially optimal item sets within budget constraints, analyzing their computational complexity.

Contribution

It introduces three preference aggregation approaches—individually best, diverse, and fair knapsack—and studies their computational complexity and tractability.

Findings

Complexity results for multiagent knapsack variants

Parameterized complexity analysis under various conditions

Insights into computational feasibility of preference-based selection

Abstract

We study the following multiagent variant of the knapsack problem. We are given a set of items, a set of voters, and a value of the budget; each item is endowed with a cost and each voter assigns to each item a certain value. The goal is to select a subset of items with the total cost not exceeding the budget, in a way that is consistent with the voters' preferences. Since the preferences of the voters over the items can vary significantly, we need a way of aggregating these preferences, in order to select the socially best valid knapsack. We study three approaches to aggregating voters' preferences, which are motivated by the literature on multiwinner elections and fair allocation. This way we introduce the concepts of individually best, diverse, and fair knapsack. We study the computational complexity (including parameterized complexity, and complexity under restricted domains) of the aforementioned multiagent variants of knapsack.