The Structural Complexity Landscape of Finding Balance-Fair Shortest Paths

TL;DR

This paper explores the parameterized complexity of finding shortest s-t paths in vertex-colored graphs with fairness constraints, providing a comprehensive classification of the problem's computational difficulty.

Contribution

It offers a complete complexity landscape for the problem, including classifications like polynomial kernels, fixed-parameter tractability, and hardness results.

Findings

Tetrachotomy classification of complexity

Identification of polynomial kernels and FPT algorithms

Hardness results including W[1]-hardness and para-NP-hardness

Abstract

We study the parameterized complexity of finding shortest s-t-paths with an additional fairness requirement. The task is to compute a shortest path in a vertex-colored graph where each color appears (roughly) equally often in the solution. We provide a complete picture of the parameterized complexity landscape of the problem with respect to structural parameters by showing a tetrachotomy including polynomial kernels, fixed-parameter tractability, XP-time algorithms (and W[1]-hardness), and para-NP-hardness.