We study maximum matching problems in temporal graphs whose underlying graph is a tree. We consider two temporal models. In a $Δ$-matching, selected time edges sharing an endpoint must have time ticks differing by at least $Δ$. In a $γ$-matching, the selected objects are blocks of $γ$ consecutive appearances of the same underlying edge. We also consider the related ordered static problem of $d$-distance matchings. We show that maximum $Δ$-matching remains NP-hard on temporal trees for every $Δ\geq 2$, even in the sparse case where each edge appears at most twice. Using a reduction between the temporal models, we obtain the analogous result for maximum $γ$-matching on temporal trees, even when each edge admits at most two $γ$-edges. We also show, via a reduction from $d$-distance matching, that maximum $γ$-matching is APX-hard even when the underlying graph is bipartite. Complementing these hardness results, we identify several tractable cases. We prove that maximum $Δ$-matching is polynomial-time solvable on temporal trees in which every edge appears exactly once, and that maximum $γ$-matching is polynomial-time solvable when each edge admits at most one $γ$-edge. We also give dynamic-programming algorithms under bounded local-use and local-sparsity assumptions, and derive polynomial-time solvability of maximum $d$-distance matching when the input bipartite graph is a tree. Finally, we prove that both maximum $Δ$-matching and maximum $γ$-matching admit polynomial-time approximation schemes on temporal trees.

翻译：我们研究了底层图为树的时间图中的最大匹配问题。考虑两种时间模型：在Δ-匹配中，共享端点的选定时间边的时间刻必须至少相差Δ；在γ-匹配中，选定对象为同一条底层边连续出现γ次构成的块。我们还考虑了相关的有序静态问题——d-距离匹配。我们证明，对于每个Δ≥2，即使在每条边最多出现两次的稀疏情况下，时间树上的最大Δ-匹配仍是NP难的。通过两种时间模型之间的归约，我们得到了类似结果：即使在每条边最多有两个γ-边的情况下，时间树上的最大γ-匹配也是NP难的。我们还通过从d-距离匹配的归约证明，即使底层图为二分图，最大γ-匹配也是APX难的。与这些困难结果互补，我们识别出若干可解情形。我们证明：在每条边恰好出现一次的时间树上，最大Δ-匹配可在多项式时间内求解；当每条边至多有一个γ-边时，最大γ-匹配可在多项式时间内求解。我们还在有界局部使用和有界局部稀疏性假设下给出了动态规划算法，并推导出输入二分图为树时最大d-距离匹配的多项式时间可解性。最后，我们证明时间树上的最大Δ-匹配和最大γ-匹配均允许多项式时间近似方案。