Restless reachability problems in temporal graphs

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 kinds 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 others. We present an algebraic algorithmic framework based on constrained multilinear sieving for solving the restless reachability problems we propose. In particular, parameterized by the length <i>k</i> of a path sought, we show that the proposed problems can be solved in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup><mml:mn>2</mml:mn> <mml:mi>k</mml:mi></mml:msup> <mml:mi>k</mml:mi> <mml:mi>m</mml:mi> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> time and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> space, where <i>n</i> is the number of vertices, <i>m</i> the number of edges, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi></mml:math> the maximum resting time of an input temporal graph. The approach can be extended to extract paths and connected subgraphs in both static and temporal graphs, thus improving the work of Björklund et al. (in Proceedings of the European symposium on algorithms, 2014) and Thejaswi et al. (Big Data 8:335-362, 2020). 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 on real-world graphs. Our implementation is efficiently engineered and highly optimized. For instance, we can solve the restless reachability problem by restricting the path length to 9 in a real-world graph dataset with over 36 million directed edges in less than one hour on a commodity desktop with a 4-core Haswell CPU.