Restless Temporal Path Parameterized Above Lower Bounds

Reachability questions are one of the most fundamental algorithmic primitives in temporal graphs -- graphs whose edge set changes over discrete time steps. A core problem here is the NP-hard Short Restless Temporal Path: given a temporal graph $\mathcal G$, two distinct vertices $s$ and $z$, and two numbers $δ$ and $k$, is there a $δ$-restless temporal $s$-$z$ path of length at most $k$? A temporal path is a path whose edges appear in chronological order and a temporal path is $δ$-restless if two consecutive path edges appear at most $δ$ time steps apart from each other. Among others, this problem has applications in neuroscience and epidemiology. While Short Restless Temporal Path is known to be computationally hard, e.g., it is NP-hard for only three time steps and W[1]-hard when parameterized by the feedback vertex number of the underlying graph, it is fixed-parameter tractable when parameterized by the path length $k$. We improve on this by showing that Short Restless Temporal Path can be solved in (randomized) $4^{k-d}|\mathcal G|^{O(1)}$ time, where $d$ is the minimum length of a temporal $s$-$z$ path.

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