Proportionally Fair Matching via Randomized Rounding

TL;DR

This paper introduces a probabilistic relaxation of the proportional fair matching problem, providing fast algorithms with constant-factor guarantees that achieve near-fairness with high probability, overcoming known computational hardness.

Contribution

It proposes a novel probabilistic fairness notion and develops simple, efficient algorithms for bipartite graphs that approximate maximum weight matchings while nearly satisfying proportional fairness.

Findings

Algorithms achieve constant-factor approximation guarantees.

The probabilistic fairness notion can be made arbitrarily close to perfect fairness.

The approach overcomes computational hardness barriers.

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 $\delta$-\textsc{ProbablyAlmostFair}, where we ensure proportionality up to a factor of at most $(1 \pm \delta)$ for some small $\delta >0$ with high probability. The violation $\delta$ 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 $\delta$-\textsc{ProbablyAlmostFair} matching.