Proportionally Fair Matching via Randomized Rounding

TL;DR

The paper tackles proportional fair matching on edge-colored bipartite graphs by introducing a $\delta$-ProbablyAlmostFair relaxation of $(\alpha,\beta)$-BalancedMatching and a fast LP-based randomized rounding via online contention resolution schemes. The main result is a $1/2$-approximation for the weighted objective that, with high probability, satisfies near-fair color representation; the analysis leverages a Doob martingale and Freedman’s concentration to bound color-class deviations. For the simpler one-sided fairness case ($\alpha=0$), the authors further show how to achieve exact $\beta$-fairness with high probability by perturbing the LP and obtain a matching whose expected weight is at least $\tfrac{1}{2}(1-\varepsilon)\mathrm{OPT}$. A brute-force method handles small instances efficiently, while the discussion highlights potential improvements via random-order techniques and extensions to general graphs, emphasizing both theoretical hardness and practical tractability in fair matching problems. The results advance fair matching by marrying LP relaxation with randomized rounding and advanced concentration tools to achieve robust, scalable guarantees.

Abstract

Given an edge-colored graph, the goal of the proportional fair matching problem is to find a maximum weight matching while ensuring proportional representation (with respect to the number of edges) of each color. The colors may correspond to demographic groups or other protected traits where we seek to ensure roughly equal representation from each group. It is known that, assuming ETH, it is impossible to approximate the problem with $\ell$ colors in time $2^{o(\ell)} n^{\mathcal{O}(1)}$ (i.e., subexponential in $\ell$) even on \emph{unweighted path graphs}. Further, even determining the existence of a non-empty matching satisfying proportionality is NP-Hard. To overcome this hardness, we relax the stringent proportional fairness constraints to a probabilistic notion. We introduce a notion we call $δ$-\textsc{ProbablyAlmostFair}, where we ensure proportionality up to a factor of at most $(1 \pm δ)$ for some small $δ>0$ with high probability. The violation $δ$ can be brought arbitrarily close to $0$ for some \emph{good} instances with large values of matching size. We propose and analyze simple and fast algorithms for bipartite graphs that achieve constant-factor approximation guarantees, and return a $δ$-\textsc{ProbablyAlmostFair} matching.

Figures (2)

• Figure 1: At $t=1$ when $v_t$ proposes to $u$ instead of $w$ and $\Tilde{F}_{v_t}=u$, then the expected change in the matching restricted to blue edges, $\mathcal{M}_{\text{blue}}$, is bounded by $2(1 - \epsilon)$.
• Figure 2: In the edge-colored graph $G$, we have $|\Delta M_t| = 1-\epsilon$ at each time $t$. Therefore, the sum $\sum_{t \in [n]} |\Delta M_t|$ is $\Theta(n)$.

Theorems & Definitions (46)

• Lemma 3.1
• Theorem 4.1
• Remark 4.2
• Theorem 4.3
• Remark 4.4
• Remark 4.5
• Lemma 5.1
• proof
• Theorem 5.2: freedman1975tail
• Remark 5.3