In a temporal graph, each edge is available at specific points in time. Such an availability point is often represented by a ''temporal edge'' that can be traversed from its tail only at a specific departure time, for arriving in its head after a specific travel time. In such a graph, the connectivity from one node to another is naturally captured by the existence of a temporal path where temporal edges can be traversed one after the other. When imposing constraints on how much time it is possible to wait at a node in-between two temporal edges, it then becomes interesting to consider temporal walks where it is allowed to visit several times the same node, possibly at different times. We study the complexity of computing minimum-cost temporal walks from a single source under waiting-time constraints in a temporal graph, and ask under which conditions this problem can be solved in linear time. Our main result is a linear time algorithm when the input temporal graph is given by its (classical) space-time representation. We use an algebraic framework for manipulating abstract costs, enabling the optimization of a large variety of criteria or even combinations of these. It allows to improve previous results for several criteria such as number of edges or overall waiting time even without waiting constraints. It saves a logarithmic factor for all criteria under waiting constraints. Interestingly, we show that a logarithmic factor in the time complexity appears to be necessary with a more basic input consisting of a single ordered list of temporal edges (sorted either by arrival times or departure times). We indeed show equivalence between the space-time representation and a representation with two ordered lists.

翻译：在时间图中，每条边在特定的时间点可用。这种可用性点通常表示为“时间边”，它只能从起点在特定出发时间被遍历，并在经过特定旅行时间后到达终点。在此类图中，从一个节点到另一个节点的连通性自然由时间路径的存在性来刻画，其中时间边可以依次被遍历。当对两个时间边之间在节点上可等待的时间施加约束时，考虑允许多次访问同一节点（可能在不同时间）的时间游走便变得有趣。我们研究了在时间图中，在等待时间约束下从单一源点计算最小成本时间游走的复杂性，并探讨了在何种条件下该问题可在线性时间内解决。我们的主要结果是：当输入时间图由其（经典）时空表示给出时，存在一个线性时间算法。我们采用一个代数框架来操作抽象成本，从而能够优化多种标准甚至它们的组合。这改进了先前针对若干标准（如边数或总等待时间）的结果，即使在没有等待约束的情况下也是如此。对于所有带有等待约束的标准，它节省了一个对数因子。有趣的是，我们表明，当输入仅由一个有序的边列表（按到达时间或出发时间排序）构成时，时间复杂性中的一个对数因子似乎是必要的。我们确实证明了时空表示与具有两个有序列表的表示之间的等价性。