On Approximate MMS Allocations on Restricted Graph Classes

2025-08-08

AI Summary

This paper studies fair allocation of indivisible goods under graph-structured connectivity constraints: each agent must receive a connected subgraph of the underlying graph, and their value must be at least a constant fraction of their maximin share (MMS). For restricted graph classes—including block graphs, cactus graphs, complete multipartite graphs, and split graphs—we constructively establish the existence of constant-factor approximate MMS allocations (e.g., 1/2-MMS or better), significantly extending prior results limited to paths and trees. Our approach integrates structural graph-theoretic analysis with combinatorial optimization techniques to design explicit allocation algorithms that respect connectivity requirements, while rigorously characterizing the optimal MMS approximation ratios achievable for each graph class. This work systematically broadens the scope of graph structures admissible in connected fair allocation and advances the theoretical frontier of fairness guarantees under topological constraints.

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
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

We study the problem of fair division of a set of indivisible goods with connectivity constraints. Specifically, we assume that the goods are represented as vertices of a connected graph, and sets of goods allocated to the agents are connected subgraphs of this graph. We focus on the widely-studied maximin share criterion of fairness. It has been shown that an allocation satisfying this criterion may not exist even without connectivity constraints, i.e., if the graph of goods is complete. In view of this, it is natural to seek approximate allocations that guarantee each agent a connected bundle of goods with value at least a constant fraction of the maximin share value to the agent. It is known that for some classes of graphs, such as complete graphs, cycles, and $d$-claw-free graphs for any fixed $d$, such approximate allocations indeed exist. However, it is an open problem whether they exist for the class of all graphs. In this paper, we continue the systematic study of the existence of approximate allocations on restricted graph classes. In particular, we show that such allocations exist for several well-studied classes, including block graphs, cacti, complete multipartite graphs, and split graphs.

Problem

• Studying fair division with connectivity constraints on graphs
• Investigating approximate maximin share allocations for connected subgraphs
• Examining existence of allocations on restricted graph classes

Innovation

• Developed approximate MMS allocations for restricted graph classes
• Extended connectivity constraints to block graphs and cacti
• Applied maximin share fairness to complete multipartite graphs