Fair Knapsack

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.

Fair Allocation of Indivisible Public Goods

“…In particular, we show that when the feasibility constraints encode independent sets of a matroid M, maximizing the smooth Nash welfare objective in Equation (2) with ℓ = 1 yields a (0, 2)-core outcome. However, optimizing this objective is known to be NP-hard [12]. We also show that given ǫ > 0, a local search procedure for this objective function (given below) yields a (0, 2 + ǫ)-core outcome in polynomial time, which proves Theorem 1.…”

“…We consider a fairly broad model for public goods allocation that generalizes much of previous work [24,10,11,3,12,9,30]. There is a set of voters (or agents) N = [n].…”

“…Packing constraints capture the general Knapsack setting, in which there is a set of m items, each item j has an associated size s j , and a set of items of total size at most B must be selected. This setting is motivated by participatory budgeting applications [29,18,17,11,15,12,6], in which the items are public projects, and the sizes represents the costs of the projects. Knapsack uses a single packing constraint.…”

“…We show that maximizing this smooth objective function achieves a good approximation to the core. However, maximizing this objective is still NP-hard [12], so we devise local search procedures that run in polynomial time and still give good approximations of the core. In effect, we make a novel connection between appropriate local optima of smooth Nash Welfare objectives and the core.…”

“…Note that ℓ = 0 coincides with the Nash welfare objective. The case of ℓ = 1 was considered by [12], who showed it is NP-Hard to optimize. Recall that we normalized agent utilities so that each agent has a maximum utility of 1 for any element, so when we add ℓ to the utility of agent i, it is equivalent to adding ℓ max j u ij to the utility of agent i when utilities are not normalized.…”