时间路径覆盖：Dilworth性质与参数化复杂性
Temporal Path Covers: Dilworth Properties and Parameterized Complexity
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中文总结 AI 辅助
研究时间有向无环图上的最小时间路径覆盖问题，证明在满足Dilworth性质时多项式时间可解，并给出参数化复杂性结果，包括W[1]-难和FPT算法。
中文摘要 AI 辅助
最小时间路径覆盖（TPC）和最小时间不相交路径覆盖（TDPC）问题由[Chakraborty, Dailly, Foucaud, Klasing, MFCS '24]提出。两者在时间DAG上被证明是NP难的，而后者在时间有向树上也是NP难的。该论文中建立的T(D)PC的所有可解情况都满足时间Dilworth性质，即最小T(D)PC的大小等于最大反链的大小。这引出一个自然问题：在承诺相应Dilworth性质成立的情况下，T(D)PC是否可以在多项式时间内求解？在这项工作中，我们对两个问题都给出了肯定回答，实际上证明了在相应承诺下，最小T(D)PC的大小恰好等于连通图的Lovász数。另一方面，我们为TPC和TDPC建立了参数化算法和难度结果。我们的主要结果是：即使对于只有两个时间步的时间图，以删除距离到线性森林为参数，TPC是W[1]-难的，这否定了Chakraborty等人关于以树宽加时间步数为参数的XP算法能否改进为FPT的开放问题。另一方面，我们证明如果使用顶点覆盖数代替上述参数化中的树宽，则存在FPT算法。我们补充证明了将时间步数包含在参数中对实现可解性是必要的，否则即使顶点覆盖大小为常数，TPC和TDPC仍然是NP难的。在此过程中，我们还建立了涉及底层图的路径宽和最大度等结构参数的各种para-NP-难结果。
英文摘要
The Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) problems were introduced by [Chakraborty, Dailly, Foucaud, Klasing, MFCS '24] as the natural temporal analogues of Minimum Path Cover and Minimum Path Partition, and were shown to be NP-hard on temporal DAGs. All tractable graph classes for T(D)PC established in that paper satisfy a temporal Dilworth property, namely that the size of the respective minimum temporal path cover is equal to the size of the maximum antichain. This raises a natural question: is T(D)PC guaranteed to be polynomial-time solvable under the promise that the respective Dilworth property holds? For both problems, we answer this question in the affirmative for their decision version and in the negative for their search version (under P $\neq$ NP). In another direction, we establish parameterized algorithms and hardness results for TPC and TDPC. Our main result is that TPC is W[1]-hard parameterized by the deletion distance to linear forest even for temporal graphs with two time-steps, answering in the negative an open question by Chakraborty et al. about whether an XP algorithm parameterized by treewidth plus number of time-steps can be improved to FPT. On the other hand, we prove that an FPT algorithm does exist if the vertex cover number is used as parameter instead of the treewidth in the above parameterization. We complement this with a proof that including the number of time-steps in the parameter is necessary to yield tractability, as, otherwise, both TPC and TDPC remain NP-hard even for constant vertex cover size. Along the way, we establish various other para-NP-hardness results involving structural parameters such as the pathwidth and the maximum degree of the underlying graph.
