Allocation of Indivisible Items with Individual Preference Graphs

TL;DR

This paper explores the allocation of indivisible items based on agents' preference graphs, introducing a dissatisfaction measure and analyzing the computational complexity of minimizing dissatisfaction in various graph structures.

Contribution

It introduces a new dissatisfaction measure for indivisible item allocation based on preference graphs and studies the complexity of minimizing dissatisfaction.

Findings

NP-hardness results for general cases

Polynomial algorithms for specific graph structures

Parameterized complexity analysis with respect to various parameters

Abstract

This paper studies the allocation of indivisible items to agents, when each agent's preferences are expressed by means of a directed acyclic graph. The vertices of each preference graph represent the subset of items approved of by the respective agent. An arc $(a,b)$ in such a graph means that the respective agent prefers item $a$ over item $b$. We introduce a new measure of dissatisfaction of an agent by counting the number of non-assigned items which are approved of by the agent and for which no more preferred item is allocated to the agent. Considering two problem variants, we seek an allocation of the items to the agents in a way that minimizes (i) the total dissatisfaction over all agents or (ii) the maximum dissatisfaction among the agents. For both optimization problems we study the status of computational complexity and obtain NP-hardness results as well as polynomial algorithms with respect to natural underlying graph structures, such as stars, trees, paths, and matchings. We also analyze the parameterized complexity of the two problems with respect to various parameters related to the number of agents, the dissatisfaction threshold, the vertex degrees of the preference graphs, and the treewidth.