Cargando…
Obtaining a Proportional Allocation by Deleting Items
We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the re...
Autores principales: | , , |
---|---|
Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer US
2021
|
Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550255/ https://www.ncbi.nlm.nih.gov/pubmed/34720294 http://dx.doi.org/10.1007/s00453-020-00794-4 |
Sumario: | We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is [Formula: see text] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation [Formula: see text] in advance as part of the input, and our aim is to delete a minimum number of items such that [Formula: see text] is proportional in the remainder; this variant turns out to be [Formula: see text] -hard for six agents, but polynomial-time solvable for two agents, and we show that it is [Formula: see text] -hard when parameterized by the number k of |
---|