Beyond the Half-Approximation: Fair and Efficient Online Class Matching

2026-05-22

AI Summary

This work addresses the challenge of surpassing the long-standing barrier that limits utilitarian social welfare (USW) to at most half of the optimal value in online bipartite matching under group fairness constraints, specifically class envy-freeness (CEF). The authors propose a threshold-based online matching algorithm that initially prioritizes equitable allocation to satisfy fairness and, upon reaching a predetermined threshold, switches to maximizing efficiency. For the first time, this approach simultaneously achieves a USW approximation ratio strictly better than 1/2 and a constant CEF guarantee in both divisible and indivisible settings: in the divisible case, it attains $(1−e^{−γ})$-CEF and $(1−e^{γ−1}/(γ+1))$-USW; in the indivisible case, it guarantees $\gamma/2$-CEF with the same USW bound. The paper also establishes novel theoretical bounds that characterize the trade-off between fairness and efficiency.

Abstract

Online bipartite matching, where agents are known in advance but items arrive sequentially and must be irrevocably assigned, is fundamental to problems ranging from ride-sharing to online advertising. When agents belong to classes such as demographic groups or geographic regions, fairness demands equitable treatment across these groups. Recent work introduced class envy-freeness (CEF), a natural extension of the classical fair division notion: an algorithm is $α$-CEF if each class receives value at least an $α$ fraction of what it could extract from any other class's bundle. However, all known algorithms achieving constant-factor CEF guarantees attain utilitarian social welfare (total matching value) of at most $\frac{1}{2}$ times the optimum, far below the $1-\frac{1}{e} \approx 0.632$ achievable without fairness constraints. We resolve the open question of whether fairness necessitates this efficiency loss, by introducing threshold-based algorithms parameterized by $γ\in [0,1]$ that equalize allocations across classes until threshold $γ$, then maximize efficiency. For divisible matching, this yields simultaneous $(1-e^{-γ})$-CEF and $(1 - \frac{e^{γ-1}}{γ+1})$-USW guarantees; for indivisible matching, $\fracγ{2}$-CEF with the same USW. Setting $γ> 0$ produces the first algorithms beating $\frac{1}{2}$-USW while maintaining constant CEF. We complement this with a novel upper bound construction, proving no non-wasteful $α$-CEF algorithm can exceed $\frac{1 +α- e^{α-1}}{1+α}$-USW and correcting prior bounds that were vacuous for $α< 0.58$. Our upper bound nearly matches our algorithms' performance, giving the first substantive characterization of the price of fairness in online class matching.

Problem

• online bipartite matching
• fairness
• class envy-freeness
• social welfare
• price of fairness

Innovation

• online bipartite matching
• class envy-freeness
• threshold-based algorithm
• social welfare
• price of fairness