Minimizing Maximum Dissatisfaction in the Allocation of Indivisible Items under a Common Preference Graph

TL;DR

The paper studies fair allocation of indivisible items among $k$ agents under a common partial order represented by a DAG $G=(V,A)$, where an agent's dissatisfaction $\delta_\pi(i)$ counts non-dominated items not allocated to them, and the goal is to minimize the maximum $\delta_\pi(i)$. It establishes a clear complexity dichotomy: the problem is solvable in polynomial time for $k=2$ but NP-hard for any $k\ge3$, even on restricted DAGs; it then identifies tractable graph classes, including width at most $2$ graphs, out-stars, and out-forests (with constant $k$), and shows fixed-parameter tractability via modular partitions into path and independent-set modules. The authors develop and combine diverse algorithmic tools—greedy strategies, dynamic programming on trees, maximum network flow, bottleneck and ordinary matchings, and integer linear programming—to map the problem's complexity landscape and provide practical algorithms. These results contribute a nuanced understanding of fair division under common, DAG-structured preferences and offer versatile techniques for related allocation problems in theoretical and applied settings.

Abstract

We consider the task of allocating indivisible items to agents, when the agents' preferences over the items are identical. The preferences are captured by means of a directed acyclic graph, with vertices representing items and an edge $(a,b)$, meaning that each of the agents prefers item $a$ over item $b$. The dissatisfaction of an agent is measured by the number of items that the agent does not receive and for which it also does not receive any more preferred item. The aim is to allocate the items to the agents in a fair way, i.e., to minimize the maximum dissatisfaction among the agents. We study the status of computational complexity of that problem and establish the following dichotomy: the problem is NP-hard for the case of at least three agents, even on fairly restricted graphs, but polynomially solvable for two agents. We also provide several polynomial-time results with respect to different underlying graph structures, such as graphs of width at most two and tree-like structures such as stars and matchings. These findings are complemented with fixed parameter tractability results related to path modules and independent set modules. Techniques employed in the paper include bottleneck assignment problem, greedy algorithm, dynamic programming, maximum network flow, and integer linear programming.

Figures (2)

• Figure 1: Illustration of a graph of width $2$.
• Figure 2: If paths have length (at least) $k$, then the natural lower bound $3(k-1)/2$ obtained by summing up all dissatisfaction values over all vertices, which gives $3k(k-1)/2$, and dividing this value by $k$, can be obtained by the above construction based on the parity of $k$.

Theorems & Definitions (19)

• Lemma 2
• proof
• Lemma 3: folklore
• Theorem 4
• proof
• Theorem 5
• proof
• Theorem 6
• proof
• Theorem 7