Temporal Graph Realization With Bounded Stretch

2025-04-19

AI Summary

This paper studies the edge-labeling problem for periodic temporal graphs: given a static graph $G$ and period $Delta$, assign time labels from ${1,dots,Delta}$ to edges so that the length of the fastest temporal path between any vertex pair is at most $alpha$ times its distance in $G$—termed the *stretch factor*. We introduce the *bounded-stretch* optimization criterion. We design a polynomial-time approximation algorithm based on graph radius, rigorously guaranteeing $alpha < Delta$, and achieving optimal stretch in several special cases. We establish a tight complexity characterization: the problem is W[2]-hard with respect to $Delta$, yet XP-time solvable parameterized by the stretch factor $alpha$. Furthermore, we provide matching (tight) approximation lower and upper bounds—i.e., an asymptotically optimal approximation ratio.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first $Delta$ time steps, and then it reappears recurrently every $Delta$ time steps, where $Delta$ is a given period length. This model offers a natural abstraction of transportation networks where each transportation link connects two destinations periodically. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are `stretched', compared to their physical distances, i.e. their distances in the underlying static (non-temporal) graph. Given a static graph $G$, the task is to assign to each edge one time-label between 1 and $Delta$ such that, in the resulting periodic temporal graph with period~$Delta$, the duration of the fastest temporal path from any vertex $u$ to any other vertex $v$ is at most $alpha$ times the distance between $u$ and $v$ in $G$. Here, the value of $alpha$ measures how much the shortest paths are allowed to be emph{stretched} once we assign the periodic time-labels. Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple polynomial-time algorithm (the extit{radius-algorithm}) which always guarantees an approximation strictly smaller than $Delta$, and which also computes the optimum stretch in some cases. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most $k$ edges; for this problem we prove that it is W[2]-hard but admits an XP algorithm with respect to $k$.

Problem

• Optimize edge time-labels in periodic temporal graphs
• Bound temporal path stretch relative to static distances
• Develop algorithms for NP-hard temporal graph realization

Innovation

• Approximation and NP-hardness results for temporal graphs
• Polynomial-time radius-algorithm for optimal stretch
• Parameterized local search with XP algorithm