The Complexity of Computing Maximin Share Allocations on Graphs

Abstract

Maximin share is a compelling notion of fairness proposed by Buddish as a relaxation of more traditional concepts for fair allocations of indivisible goods. In this paper we consider this notion within a setting where bundles of goods must induce connected subsets over an underlying graph. This setting received much attention in earlier literature, and our study answers a number of questions that were left open. First, we show that computing maximin share allocations is FΔ2P-complete, even when focusing on consistent scenarios, that is, where such allocations are a-priori guaranteed to exist. Moreover, the problem remains intractable if all agents have the same type, i.e., have the same utility functions, and if either the values returned by the utility functions are polynomially bounded, or the underlying graphs have a low degree of cyclicity (more precisely, have bounded treewidth). However, if these conditions hold all together, then computing maximin share allocations (or checking that none exists) becomes tractable. The result is established via machineries based on logspace alternating machines that use partial representations of connected bundles, which are interesting in their own.

A Little Charity Guarantees Fair Connected Graph Partitioning

“…While envy-freeness and proportionality can be achieved exactly when the goods are divisible, as in cake-cutting (Stromquist 1980;Su 1999), as illustrated in the introduction, not even a reasonable approximation of these guarantees can be provided when the goods are indivisible, modeled as nodes of a graph. Hence, this literature focuses on special families of graphs, such as path graphs, for which such guarantees can be provided (Bouveret et al 2017;Bilò et al 2019;Bei et al 2021a), and on the computational complexity of the existence of fair connected allocations (Deligkas et al 2021;Greco and Scarcello 2020;Igarashi and Peters 2019). Our goal is to provide approximate fairness guarantees for general graphs, by using the idea of charity, which has been explored recently for fair division without the connectedness constraint (Chaudhury et al 2021b;Caragiannis, Gravin, and Huang 2019;Chaudhury et al 2021a;Berger et al 2021).…”

Guaranteeing Half-Maximin Shares Under Cardinality Constraints

“…Another, studied by Bouveret et al, uses an underlying graph to represent connectivity between the items and requires each bundle to form a connected component [5]. Such connectivity constraints have since been explored in many papers [e.g., 3,18,23]. A variation is allocation of conflicting items, where each bundle must be an independent set in the graph [10,20].…”

How to Cut a Discrete Cake Fairly

“…Bouveret et al (2017) proposed a model for allocating indivisible goods under connectivity constraints of a graph. The graph fair division has attracted a great deal of attention since then (Bouveret, Cechlárová, and Lesca 2019;Igarashi and Peters 2019;Bei et al 2021;Truszczynski and Lonc 2020;Bilò et al 2019;Bouveret et al 2017;Greco and Scarcello 2020;Deligkas et al 2021). Bilò et al (2019) developed several methods to obtain an EF1 division under connectivity constraints of a path when the number of agents is two, three, or four, or when the agents have identical monotone valuations.…”