Fairness in Graph-Theoretical Optimization Problems

TL;DR

This paper explores fairness in graph-theoretic optimization, analyzing measures for individual and group fairness, revealing their computational complexity, and connecting them to hypergraph parameters.

Contribution

It introduces and analyzes two novel individual-fairness measures based on probability distributions, linking them to hypergraph parameters and extending to group fairness.

Findings

Computing fairness measures is equivalent to hypergraph fractional covering and partitioning.

For independence systems, the two fairness measures coincide.

Determines the computational complexity of group-fair solutions for matching.

Abstract

There is arbitrariness in optimum solutions of graph-theoretic problems that can give rise to unfairness. Incorporating fairness in such problems, however, can be done in multiple ways. For instance, fairness can be defined on an individual level, for individual vertices or edges of a given graph, or on a group level. In this work, we analyze in detail two individual-fairness measures that are based on finding a probability distribution over the set of solutions. One measure guarantees uniform fairness, i.e., entities have equal chance of being part of the solution when sampling from this probability distribution. The other measure maximizes the minimum probability for every entity of being selected in a solution. In particular, we reveal that computing these individual-fairness measures is in fact equivalent to computing the fractional covering number and the fractional partitioning number of a hypergraph. In addition, we show that for a general class of problems that we classify as independence systems, these two measures coincide. We also analyze group fairness and how this can be combined with the individual-fairness measures. Finally, we establish the computational complexity of determining group-fair solutions for matching.