On the Price of Fairness of Allocating Contiguous Blocks

TL;DR

This paper investigates the unavoidable welfare loss when allocating indivisible items on a line fairly with connectedness constraints, focusing on relaxations of fairness notions and providing tight bounds with efficient algorithms.

Contribution

It introduces tight bounds on the price of fairness for connected allocations using relaxations of maximin share and proportionality, with polynomial-time algorithms.

Findings

Almost tight ratios of PoF are established for various fairness relaxations.

Polynomial-time algorithms are designed to compute allocations meeting the bounds.

The study extends fairness analysis to connected allocations for indivisible items.

Abstract

In this work, we revisit the problem of fairly allocating a number of indivisible items that are located on a line to multiple agents. A feasible allocation requires that the allocated items to each agent are connected on the line. The items can be goods on which agents have non-negative utilities, or chores on which the utilities are non-positive. Our objective is to understand the extent to which welfare is inevitably sacrificed by enforcing the allocations to be fair, i.e., price of fairness (PoF). We study both egalitarian and utilitarian welfare. Previous works by Suksompong [Discret. Appl. Math., 2019] and H\"ohne and van Stee [Inf. Comput., 2021] have studied PoF regarding the notions of envy-freeness and proportionality. However, these fair allocations barely exist for indivisible items, and thus in this work, we focus on the relaxations of maximin share fairness and proportionality up to one item, which are guaranteed to be satisfiable. For most settings, we give (almost) tight ratios of PoF and all the upper bounds are proved by designing polynomial time algorithms.