In this paper we initiate the study of the \emph{temporal graph realization} problem with respect to the fastest path durations among its vertices, while we focus on periodic temporal graphs. Given an $n \times n$ matrix $D$ and a $\Delta \in \mathbb{N}$, the goal is to construct a $\Delta$-periodic temporal graph with $n$ vertices such that the duration of a \emph{fastest path} from $v_i$ to $v_j$ is equal to $D_{i,j}$, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs has been well studied and understood since the 1960's, and this area of research remains active until nowadays. As it turns out, the periodic temporal graph realization problem has a very different computational complexity behavior than its static (i.e. non-temporal) counterpart. First we show that the problem is NP-hard in general, but polynomial-time solvable if the so-called underlying graph is a tree or a cycle. Building upon those results, we investigate its parameterized computational complexity with respect to structural parameters of the underlying static graph which measure the ``tree-likeness''. For those parameters, we essentially give a tight classification between parameters that allow for tractability (in the FPT sense) and parameters that presumably do not. We show that our problem is W[1]-hard when parameterized by the \emph{feedback vertex number} of the underlying graph, while we show that it is in FPT when parameterized by the \emph{feedback edge number} of the underlying graph.

翻译：本文首次研究了针对顶点间最快路径持续时间的时序图实现问题，重点聚焦于周期性时序图。给定一个$n \times n$矩阵$D$和$\Delta \in \mathbb{N}$，目标在于构建一个包含$n$个顶点的$\Delta$-周期时序图，使得从$v_i$到$v_j$的\textit{最快路径}持续时间等于$D_{i,j}$，或判定这样的时序图不存在。该问题在静态图上的变体自20世纪60年代以来已被充分研究并理解，且该研究领域至今仍保持活跃。结果表明，周期性时序图实现问题与其静态（即非时序）对应问题在计算复杂性上存在显著差异。我们首先证明该问题在一般情况下是NP困难的，但当所谓的底层图为树或环时可在多项式时间内求解。基于这些结果，我们研究了该问题在底层静态图结构参数（用于度量“树形程度”）下的参数化计算复杂性。针对这些参数，我们给出了一个严格分类：一类参数允许可解性（在FPT意义上），另一类参数则可能不允许。我们证明当参数化为底层图的\textit{反馈顶点数}时该问题为W[1]-难，而当参数化为底层图的\textit{反馈边数}时该问题属于FPT。