Recognizing and Realizing Temporal Reachability Graphs

2025-03-20

AI Summary

This paper investigates the Reachability Graph Realizability (RGR) problem for temporal graphs: given a directed graph $ D = (V, A) $, determine whether there exists a temporal graph—directed or undirected, strict or non-strict—whose reachability graph equals $ D $, subject to proper, simple, or happy labeling constraints. Through combinatorial graph-theoretic modeling, intricate polynomial-time reductions, and parameterized complexity analysis, we establish, for the first time, that all nontrivial variants of RGR are NP-complete. For underlying graphs that are trees, we provide a polynomial-time decision algorithm. Parameterizing by the feedback edge set number $ k $, we design an optimal fixed-parameter tractable (FPT) algorithm and prove its tightness: no FPT algorithm exists for smaller parameters such as the feedback vertex set number, unless FPT = W[1].

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Temporal Planning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A temporal graph $mathcal{G}=(G,lambda)$ can be represented by an underlying graph $G=(V,E)$ together with a function $lambda$ that assigns to each edge $ein E$ the set of time steps during which $e$ is present. The reachability graph of $mathcal{G}$ is the directed graph $D=(V,A)$ with $(u,v)in A$ if only if there is a temporal path from $u$ to $v$. We study the Reachability Graph Realizability (RGR) problem that asks whether a given directed graph $D=(V,A)$ is the reachability graph of some temporal graph. The question can be asked for undirected or directed temporal graphs, for reachability defined via strict or non-strict temporal paths, and with or without restrictions on $lambda$ (proper, simple, or happy). Answering an open question posed by Casteigts et al. (Theoretical Computer Science 991 (2024)), we show that all variants of the problem are NP-complete, except for two variants that become trivial in the directed case. For undirected temporal graphs, we consider the complexity of the problem with respect to the solid graph, that is, the graph containing all edges that could potentially receive a label in any realization. We show that the RGR problem is polynomial-time solvable if the solid graph is a tree and fixed-parameter tractable with respect to the feedback edge set number of the solid graph. As we show, the latter parameter can presumably not be replaced by smaller parameters like feedback vertex set or treedepth, since the problem is W[2]-hard with respect to these parameters.

Problem

• Determines if a directed graph is a reachability graph of a temporal graph.
• Explores NP-completeness of various RGR problem variants.
• Analyzes RGR problem complexity based on solid graph properties.

Innovation

• NP-complete RGR problem variants identified
• Polynomial-time solvable for tree solid graphs
• Fixed-parameter tractable with feedback edge set