Structural parameterizations of Geodetic Set on directed (acyclic) graphs

2026-06-24

AI Summary

This study addresses the parameterized complexity of the minimum geodetic set problem in directed graphs: given a digraph, find the smallest vertex set such that every vertex lies on a shortest directed path between some pair of vertices in the set. Leveraging parameterized complexity theory, kernelization techniques, and the Exponential Time Hypothesis (ETH), the work introduces the first algorithmic framework parameterized by structural measures such as vertex cover number (vcn) and reachable diameter (rdiam). Key contributions include a kernel of size 2^{O(vcn)} and an algorithm running in 2^{O(vcn²)}·n^{O(1)} time for general digraphs; a kernel of size (kΔ)^{O(rdiam)} for the combined parameter (k, Δ, rdiam), shown to be conditionally optimal; and several hardness results establishing W[2]-hardness and para-NP-hardness, demonstrating that no single parameter alone suffices for efficient solvability.

Abstract

In DIRECTED GEODETIC SET, we are given a (directed) graph and seek a small solution set $S \subseteq V(G)$ such that every vertex lies on a shortest directed path between two vertices in $S$. It is known that the problem is W[2]-hard when parameterized by the solution size $k$, even on directed acyclic graphs (DAGs). Our first result is a kernel of size $2^{O(vcn)}$ for DIRECTED GEODETIC SET on general digraphs, where $vcn$ denotes the vertex cover number of the underlying (undirected) graph. This implies an algorithm running in time $2^{O(vcn^2)} \cdot n^{O(1)}$. Furthermore, we prove that, assuming the ETH, the problem does not admit an algorithm running in time $2^{o(vcn^2)} \cdot n^{O(1)}$. Next, we show that on general digraphs, DIRECTED GEODETIC SET admits a natural kernel of size $(kΔ)^{O(rdiam)}$, where $Δ$ is the maximum degree and $rdiam$ denotes the reachability diameter of the digraph (a natural analogue of diameter of undirected graphs). This yields an algorithm running in time $(kΔ)^{O(rdiam \cdot k)}\cdot n^{O(1)}$. We further prove that, assuming the ETH, the problem does not admit an algorithm running in time $(kΔ)^{o(rdiam \cdot k)} \cdot n^{O(1)}$. Finally, we justify the necessity of combining parameters by establishing the following hardness results for DIRECTED GEODETIC SET: - It is W[2]-hard parameterized by $k$, even on digraphs of maximum degree 3. - It is para-NP-hard parameterized by maximum degree and reachability diameter. One can infer that the problem remains W[2]-hard when parameterized by k, even on graphs of reachability diameter 3 from Araújo and Arraes [DAM 2022]. All our conditional lower bounds and hardness results hold even when the input digraph is restricted to be a DAG.

Problem

• Directed Geodetic Set
• parameterized complexity
• vertex cover number
• reachability diameter
• W[2]-hardness

Innovation

• parameterized complexity
• directed geodetic set
• kernelization
• reachability diameter
• vertex cover number