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Restless reachability problems in temporal graphs

Suhas Thejaswi, Juho Lauri, Aristides Gionis

TL;DR

The paper tackles restless reachability in temporal graphs with waiting-time constraints and vertex colors, formalizing five problem families (RestlessPath, k-RestlessPath, RestlessReach, k-RestlessReach, and k-RestlessMotif/Reach). It introduces an algebraic, fingerprinting-based framework using constrained multilinear sieving that yields randomized decision oracles with time $O(2^k k m \Delta)$ and space $O(n\Delta)$, and further provides a fine-grained oracle for extracting actual solutions and handling color constraints. The authors prove conditional optimality under the Set Cover Conjecture and demonstrate scalability to graphs with up to $10^9$ temporal edges via open-source implementations and extensive experiments, including a case study on immunization strategies. The work advances exact algorithms for NP-hard temporal-graph problems and offers practical tools for epidemiology, neuroscience, and travel planning, while outlining avenues for handling larger path lengths and leveraging vector/GPU architectures in future work.

Abstract

We study a family of reachability problems under waiting-time restrictions in temporal and vertex-colored temporal graphs. Given a temporal graph and a set of source vertices, we find the set of vertices that are reachable from a source via a time-respecting path, where the difference in timestamps between consecutive edges is at most a resting time. Given a vertex-colored temporal graph and a multiset query of colors, we find the set of vertices reachable from a source via a time-respecting path such that the vertex colors of the path agree with the multiset query and the difference in timestamps between consecutive edges is at most a resting time. These kind of problems have applications in understanding the spread of a disease in a network, tracing contacts in epidemic outbreaks, finding signaling pathways in the brain network, and recommending tours for tourists, among other. We present an algebraic algorithmic framework based on constrained multi\-linear sieving for solving the restless reachability problems we propose. In particular, parameterized by the length $k$ of a path sought, we show that the proposed problems can be solved in $O(2^k k m Δ)$ time and $O(n Δ)$ space, where $n$ is the number of vertices, $m$ the number of edges, and $Δ$ the maximum resting time of an input temporal graph. In addition, we prove that our algorithms for the restless reachability problems in vertex-colored temporal graphs are optimal under plausible complexity-theoretic assumptions. Finally, with an open-source implementation, we demonstrate that our algorithm scales to large graphs with up to one billion temporal edges, despite the problems being NP-hard. Specifically, we present extensive experiments to evaluate our scalability claims both on synthetic and real-world graphs. Our implementation is efficiently engineered and highly optimized.

Restless reachability problems in temporal graphs

TL;DR

The paper tackles restless reachability in temporal graphs with waiting-time constraints and vertex colors, formalizing five problem families (RestlessPath, k-RestlessPath, RestlessReach, k-RestlessReach, and k-RestlessMotif/Reach). It introduces an algebraic, fingerprinting-based framework using constrained multilinear sieving that yields randomized decision oracles with time and space , and further provides a fine-grained oracle for extracting actual solutions and handling color constraints. The authors prove conditional optimality under the Set Cover Conjecture and demonstrate scalability to graphs with up to temporal edges via open-source implementations and extensive experiments, including a case study on immunization strategies. The work advances exact algorithms for NP-hard temporal-graph problems and offers practical tools for epidemiology, neuroscience, and travel planning, while outlining avenues for handling larger path lengths and leveraging vector/GPU architectures in future work.

Abstract

We study a family of reachability problems under waiting-time restrictions in temporal and vertex-colored temporal graphs. Given a temporal graph and a set of source vertices, we find the set of vertices that are reachable from a source via a time-respecting path, where the difference in timestamps between consecutive edges is at most a resting time. Given a vertex-colored temporal graph and a multiset query of colors, we find the set of vertices reachable from a source via a time-respecting path such that the vertex colors of the path agree with the multiset query and the difference in timestamps between consecutive edges is at most a resting time. These kind of problems have applications in understanding the spread of a disease in a network, tracing contacts in epidemic outbreaks, finding signaling pathways in the brain network, and recommending tours for tourists, among other. We present an algebraic algorithmic framework based on constrained multi\-linear sieving for solving the restless reachability problems we propose. In particular, parameterized by the length of a path sought, we show that the proposed problems can be solved in time and space, where is the number of vertices, the number of edges, and the maximum resting time of an input temporal graph. In addition, we prove that our algorithms for the restless reachability problems in vertex-colored temporal graphs are optimal under plausible complexity-theoretic assumptions. Finally, with an open-source implementation, we demonstrate that our algorithm scales to large graphs with up to one billion temporal edges, despite the problems being NP-hard. Specifically, we present extensive experiments to evaluate our scalability claims both on synthetic and real-world graphs. Our implementation is efficiently engineered and highly optimized.

Paper Structure

This paper contains 28 sections, 13 theorems, 11 equations, 20 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Terminology
  4. Restless path and restless reachability problems
  5. Algorithm
  6. A birds-eye view of our approach
  7. An overview of the algorithm
  8. Generating restless walks
  9. Introducing vertex-color constraints
  10. A fine-grained decision oracle
  11. Discussion
  12. Extracting an optimal solution using $k$ queries
  13. A deterministic $\mathcal{O}\xspace^*(2^n)$ algorithm for solving $\text{\sc Restless\-Path}$
  14. Infeasibility of a $\mathcal{O}\xspace^*((2-\epsilon)^k)$-time algorithm for $k\text{\sc-Restless\-Motif}$
  15. Experiments
  16. ...and 13 more sections

Key Result

Lemma 1

The polynomial encoding $\chi_{{u},{\ell},{i}}(\vec{x},\vec{y})\xspace$ presented in Equation (eq:poly-enc:1) contains a multilinear monomial of degree $2\ell-1$ if and only if there exists a restless path of length $\ell-1$ ending at vertex $u$ reaching at time $i$.

Figures (20)

  • Figure 1: Illustration of restless path problems. A temporal graph on vertices $V=\{v_1,\dots,v_7\}$ with source $s=v_1$, destination $d=v_6$ and resting time of vertices $\delta(v_1)=\dots=\delta(v_7)=2$. Arrows represent the direction of edges and the integer value on each edge corresponds to its timestamp. On the left, an example of a restless path from $v_1$ to $v_6$ of length $5$. On the center, an example of a restless path from $v_1$ to $v_6$ when the length of the path is restricted to $4$ i.e., $k=5$. On the right, an example of a short restless (path) motif from $v_1$ to $v_6$ such that the vertex colors of the path agree with the multiset of colors in $M$. Restless paths are highlighted in bold (blue).
  • Figure 2: An illustration of generating restless walks.
  • Figure 3: An example to illustrate the fine-grained evaluation scheme. The resting time of vertices is $\delta(v_1) = \dots = \delta(v_5) = 2$. A restless path of length $3$ (i.e, $\ell=4$) from vertex $v_1$ to vertex $v_5$ is highlighted in bold.
  • Figure 4: An illustration of graph construction to solve $k\text{\sc-Restless\-Motif\-Reach}$. An instance of $k\text{\sc-Restless\-Motif\-Reach}$ with a multiset of colors (left) and the transformation of the graph and the multiset (right). For illustrative purpose, we denote the vertices with color red, black and green using diamond, circular and square shapes, respectively.
  • Figure 5: The transformation of $k \text{\sc -Restless\-Reach}$ instance (left) to $k\text{\sc-Restless\-Motif\-Reach}$ instance (right). For illustrative purposes we denote vertices with color red and black using diamond and circular shapes, respectively.
  • ...and 15 more figures

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Lemma 2: Multilinear sieving bjorklund2014determinant
  • Lemma 3
  • proof
  • Lemma 4: Constrained multilinear sieving bjorklund2016constrained
  • Lemma 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 10 more