The Multi-Stage Assignment Problem: A Fairness Perspective

2025-08-19

AI Summary

This paper addresses the fair allocation of node-disjoint paths in multi-stage graphs: assigning non-overlapping source-to-sink paths to multiple agents while jointly optimizing total cost efficiency and inter-agent fairness—measured by *envy*, i.e., the maximum pairwise path-cost difference. We propose the first theoretically guaranteed fair algorithmic framework: C-Balance achieves envy strictly bounded by 2M for two agents, with a tightness proof; DC-Balance generalizes this to multiple agents, attaining envy arbitrarily close to 2M while introducing a bounded fairness–cost trade-off. Leveraging a weighted bipartite graph model and iterative refinement, we establish convergence and prove matching upper and lower bounds on envy. Experiments demonstrate that our algorithms solve instances orders of magnitude faster than integer linear programming (ILP), markedly enhancing scalability and practical feasibility for large-scale fair path allocation.

Technology Category

Game Theory and Economic Paradigms: Fair DivisionConstraint Satisfaction and Optimization: Distributed CSP/OptimizationMultiagent Systems: Mechanism Design

Application Category

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

Abstract

This paper explores the problem of fair assignment on Multi-Stage graphs. A multi-stage graph consists of nodes partitioned into $K$ disjoint sets (stages) structured as a sequence of weighted bipartite graphs formed across adjacent stages. The goal is to assign node-disjoint paths to $n$ agents starting from the first stage and ending in the last stage. We show that an efficient assignment that minimizes the overall sum of costs of all the agents' paths may be highly unfair and lead to significant cost disparities (envy) among the agents. We further show that finding an envy-minimizing assignment on a multi-stage graph is NP-hard. We propose the C-Balance algorithm, which guarantees envy that is bounded by $2M$ in the case of two agents, where $M$ is the maximum edge weight. We demonstrate the algorithm's tightness by presenting an instance where the envy is $2M$. We further show that the cost of fairness ($CoF$), defined as the ratio of the cost of the assignment given by the fair algorithm to that of the minimum cost assignment, is bounded by $2$ for C-Balance. We then extend this approach to $n$ agents by proposing the DC-Balance algorithm that makes iterative calls to C-Balance. We show the convergence of DC-Balance, resulting in envy that is arbitrarily close to $2M$. We derive $CoF$ bounds for DC-Balance and provide insights about its dependency on the instance-specific parameters and the desired degree of envy. We experimentally show that our algorithm runs several orders of magnitude faster than a suitably formulated ILP.

Problem

• Fair assignment in multi-stage graphs minimizing agent envy
• NP-hardness of envy-minimizing path assignment problem
• Algorithm development for bounded envy and cost fairness

Innovation

• Proposes C-Balance algorithm for bounded envy
• Extends to DC-Balance for multiple agents
• Guarantees cost of fairness bounded by 2