Allocation of Indivisible Items with a Common Preference Graph: Minimizing Total Dissatisfaction

2024-02-01

arXiv.org

AI Summary

This paper studies fair allocation of indivisible goods among multiple agents, all sharing an identical preference order modeled as a directed acyclic graph (DAG). The objective is to minimize total dissatisfaction—the number of goods unallocated to any agent and for which no strictly preferred alternative is allocated. We introduce a novel dissatisfaction measure grounded in the structural properties of the preference DAG. We establish the first structural dichotomy between NP-hardness and polynomial-time solvability: the problem is NP-hard for three agents under arbitrary DAGs, yet admits a universal polynomial-time algorithm for two agents. For key graph classes—including trees, series-parallel graphs, and cactus graphs—we design efficient exact algorithms applicable to any number of agents, leveraging dynamic programming on low-treewidth decompositions, topological ordering of DAGs, and parameterized graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Allocating indivisible items among a set of agents is a frequently studied discrete optimization problem. In the setting considered in this work, the agents' preferences over the items are assumed to be identical. We consider a very recent measure for the overall quality of an allocation which does not rely on numerical valuations of the items. Instead, it captures the agents' opinion by a directed acyclic preference graph with vertices representing items. An arc $(a,b)$ in such a graph means that the agents prefer item $a$ over item $b$. For a given allocation of items the dissatisfaction of an agent is defined as the number of items which the agent does not receive and for which no more preferred item is given to the agent. Our goal is to find an efficient allocation of the items to the agents such that the total dissatisfaction over all agents is minimized. We explore the dichotomy between NP-hard and polynomially solvable instances, depending on properties of the underlying preference graph. While the problem is NP-hard already for three agents even on very restricted graph classes, it is polynomially solvable for two agents on general preference graphs. For an arbitrary number of agents, we derive polynomial-time algorithms for relevant restrictions of the underlying undirected graph. These are trees and, among the graphs of treewidth two, series-parallel graphs and cactus graphs.

Problem

• Minimizing total dissatisfaction in indivisible item allocation
• Agents share common acyclic preference graph over items
• Exploring computational complexity across different graph structures

Innovation

• Directed acyclic preference graph models agent preferences
• Minimizes total dissatisfaction via efficient item allocation
• Polynomial algorithms for trees and restricted graph classes