The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints
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Computing a (shortest) path between two vertices in a graph is one of the most fundamental primitive in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set remains static but the edge set may change over time, gained more and more attention. In a nutshell, temporal paths have to respect time, that is, they may only move forward in time. More formally, the time edges used by a temporal path either need to have increasing or non-decreasing time stamps. In is well known that computing temporal paths is polynomial-time solvable. We study a natural variant, where temporal paths may only dwell a certain given amount of time steps in any vertex, which we call restless temporal paths. This small modification creates a significant change in the computational complexity of the task of finding temporal paths. We show that finding restless temporal paths is NP-complete and give a thorough analysis of the (parameterized) computational complexity of this problem. In particular, we show that problem remains computationally hard on temporal graphs with three layers and is W[1]-hard when parameterized by the feedback vertex number of the underlying graph. On the positive side, we give an efficient (FPT) algorithm to find short restless temporal paths that has an asymptotically optimal running time assuming the Exponential Time Hypothesis.
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