Fairness in graph-theoretical optimization problems

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 a variant of the matching problem.