Papers › Restless reachability problems in temporal graphs
Restless reachability problems in temporal graphs
Suhas Thejaswi, Juho Lauri, Aristides Gionis
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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(2ᵏ 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.
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