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Beer Path Problems in Temporal Graphs

Andrea D'Ascenzo, Giuseppe F. Italiano, Sotiris Kanellopoulos, Anna Mpanti, Aris Pagourtzis, Christos Pergaminelis

TL;DR

The paper addresses computing temporal beer paths—path variants that must visit at least one beer vertex during active times—in temporal graphs with edges having start times and travel durations. It defines four objectives (EABP, LDBP, FBP, SBP) and provides two representations (edge-stream and adjacency-list) with algorithms that match the efficiency of standard temporal-path methods. It contributes both algorithmic solutions (including multisource earliest-arrival and reverse-time latest-departure and non-dominated-path techniques) and preprocessing strategies (non-dominated path precomputation and a Path_Wu-style graph transformation to a static DAG) to enable fast queries under dynamic shop-openings/closings. The work blends theoretical algorithm design with practical preprocessing tools, offering a comprehensive framework for temporal beer paths in real-world settings such as transportation and locate-visiting scenarios.

Abstract

Computing paths in graph structures is a fundamental operation in a wide range of applications, from transportation networks to data analysis. The beer path problem, which captures the option of visiting points of interest, such as gas stations or convenience stops, prior to reaching the final destination, has been recently introduced and extensively studied in static graphs. However, existing approaches do not account for temporal information, which is often crucial in real-world scenarios. For instance, transit services may follow fixed schedules, and shops may only be accessible during certain hours. In this work, we introduce the notion of beer paths in temporal graphs, where edges are time-dependent and certain vertices (beer vertices) are active only at specific time instances. We formally define the problems of computing earliest-arrival, latest-departure, fastest, and shortest temporal beer paths and propose efficient algorithms for these problems under both edge stream and adjacency list representations. The time complexity of each of our algorithms is aligned with that of corresponding temporal pathfinding algorithms, thus preserving efficiency. Additionally, we present preprocessing techniques that enable efficient query answering under dynamic conditions, for example new openings or closings of shops. We achieve this through appropriate precomputation of selected paths or by transforming a temporal graph into an equivalent static graph.

Beer Path Problems in Temporal Graphs

TL;DR

The paper addresses computing temporal beer paths—path variants that must visit at least one beer vertex during active times—in temporal graphs with edges having start times and travel durations. It defines four objectives (EABP, LDBP, FBP, SBP) and provides two representations (edge-stream and adjacency-list) with algorithms that match the efficiency of standard temporal-path methods. It contributes both algorithmic solutions (including multisource earliest-arrival and reverse-time latest-departure and non-dominated-path techniques) and preprocessing strategies (non-dominated path precomputation and a Path_Wu-style graph transformation to a static DAG) to enable fast queries under dynamic shop-openings/closings. The work blends theoretical algorithm design with practical preprocessing tools, offering a comprehensive framework for temporal beer paths in real-world settings such as transportation and locate-visiting scenarios.

Abstract

Computing paths in graph structures is a fundamental operation in a wide range of applications, from transportation networks to data analysis. The beer path problem, which captures the option of visiting points of interest, such as gas stations or convenience stops, prior to reaching the final destination, has been recently introduced and extensively studied in static graphs. However, existing approaches do not account for temporal information, which is often crucial in real-world scenarios. For instance, transit services may follow fixed schedules, and shops may only be accessible during certain hours. In this work, we introduce the notion of beer paths in temporal graphs, where edges are time-dependent and certain vertices (beer vertices) are active only at specific time instances. We formally define the problems of computing earliest-arrival, latest-departure, fastest, and shortest temporal beer paths and propose efficient algorithms for these problems under both edge stream and adjacency list representations. The time complexity of each of our algorithms is aligned with that of corresponding temporal pathfinding algorithms, thus preserving efficiency. Additionally, we present preprocessing techniques that enable efficient query answering under dynamic conditions, for example new openings or closings of shops. We achieve this through appropriate precomputation of selected paths or by transforming a temporal graph into an equivalent static graph.

Paper Structure

This paper contains 24 sections, 12 theorems, 2 figures, 14 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contribution
  4. Preliminaries and problem definition
  5. Preliminaries
  6. The temporal beer path problem
  7. Algorithms for temporal beer path problems
  8. Algorithms for earliest-arrival beer path
  9. An earliest-arrival beer path algorithm in edge stream representation
  10. A faster earliest-arrival beer path algorithm with adjacency list representation
  11. Algorithms for latest-departure beer path
  12. Algorithm for fastest beer path
  13. Algorithm for shortest beer path
  14. Beer path queries with preprocessing
  15. EABP and LDBP with precomputation of non-dominated paths
  16. ...and 9 more sections

Key Result

Lemma 9

For all nodes $x,y \in V$, if there exists some temporal $x-y$ path, then there exists some earliest-arrival $x-y$ path $\langle v_0=x,v_1,v_2,\ldots,v_k=y \rangle$ for which every prefix-subpath $\langle x,v_1,v_2,\ldots,v_i \rangle$ is an earliest-arrival $x-v_i$ path.

Figures (2)

  • Figure 1: Example in which the $x-y$ EABP/LDBP/FBP/SBP use $b_1, \ldots, b_4$ respectively. Numbers in parentheses denote $(t,\lambda)$ of edges and numbers in brackets denote beer vertex active times. $x-b_5-y$ is not a valid temporal beer path because $4\notin T_{b_5}$. The $x-b_2$ edge with $(t,\lambda)=(18,9)$ is dominated and can be removed from the graph without affecting any objective.
  • Figure 2: Graph transformation, from $G$ in (a) to $\tilde{G}$ in (b).

Theorems & Definitions (21)

  • Definition 1: Dominated edge
  • Definition 2: Dominated paths
  • Definition 3: EABP
  • Definition 4: LDBP
  • Definition 5: FBP
  • Definition 6: SBP
  • Definition 7: MSEAP
  • Lemma 9: Wu et al. Path_Wu
  • Lemma 11
  • Lemma 12
  • ...and 11 more