Fair Short Paths in Vertex-Colored Graphs

TL;DR

This paper introduces a framework for finding fair shortest paths in vertex-colored graphs, addressing diverse fairness constraints, proving computational hardness, and providing fixed-parameter tractability results.

Contribution

It models multiple fairness aspects in shortest path problems with color constraints and analyzes their computational complexity, offering new algorithmic insights.

Findings

Problems are NP-hard even with strict color frequency constraints

Fixed-parameter tractability results for path length

Intractability persists in restricted graph settings

Abstract

The computation of short paths in graphs with arc lengths is a pillar of graph algorithmics and network science. In a more diverse world, however, not every short path is equally valuable. For the setting where each vertex is assigned to a group (color), we provide a framework to model multiple natural fairness aspects. We seek to find short paths in which the number of occurrences of each color is within some given lower and upper bounds. Among other results, we prove the introduced problems to be computationally intractable (NP-hard and parameterized hard with respect to the number of colors) even in very restricted settings (such as each color should appear with exactly the same frequency), while also presenting an encouraging algorithmic result ("fixed-parameter tractability") related to the length of the sought solution path for the general problem.