The Structural Complexity Landscape of Finding Balance-Fair Shortest Paths

TL;DR

Balance-fair Shortest Path asks for an $s$-$t$-path that is both shortest and balance-fair, i.e., the color counts along the path satisfy ${\max_i |\chi_P^i| - \min_i |\chi_P^i| \le 1}$ in a vertex-colored graph. The authors provide a complete tetrachotomy across structural parameters, distinguishing polynomial kernels, fixed-parameter tractability, XP-time algorithms, and para-$\mathsf{NP}$-hardness. They develop polynomial kernels for distance to cographs and to $P_h$-free graphs, a kernel for neighborhood diversity, and show hardness results for parameters such as treedepth, minimum clique cover, and maximum leaf number via OR-cross-composition and reductions. The results reveal that fairness constraints significantly impact the complexity of a fundamental graph problem across many graph classes, guiding both theoretical understanding and practical algorithm design for fair routing problems.

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.

Figures (5)

• Figure 1: A graph with colored vertices (blue, green, and gray). The highlighted path is a shortest path between $s$ and $t$ and contains three vertices of each color. Thus, it is balance-fair.
• Figure 2: The relations between structural graph parameters and our respective results for Balance-fair Shortest Path. An edge from a parameter $\alpha$ to a parameter $\beta$ below $\alpha$ means that $\alpha$ upper-bounds $\beta$ (the bounds are usually linear functions; see also Sch19). A parameter $k$ is marked green () if Balance-fair Shortest Path admits a polynomial kernel with $k$, yellow () if it is FPT with $k$ but presumably does not admit a polynomial kernel, orange () if the respective parameterized problem is in XP and $\operatorname{W[1]}$-hard, and red () if Balance-fair Shortest Path is $\operatorname{NP}$-hard for constant $k$. Parameters without a thick border obtain their classification from parameters above or below.
• Figure 3: The vertex gadget $G_v$ for a vertex $v$ and a legend providing the names of each color.
• Figure 4: The vertex gadget $G_e$ for an edge $e = \{u,v\}$ and a legend providing color names.
• Figure 5: An extract from the filler gadget $G_f$.

Theorems & Definitions (21)

• Definition 1
• Theorem 2: fomin2016representative
• Definition 3: OR-cross-composition bodlaender2014kernel
• proof
• proof
• proof
• Proposition 9
• proof
• Theorem 10
• proof