Temporal Path Covers: Dilworth Properties and Parameterized Complexity

2026-06-30

AI Summary

This study investigates the computational complexity of minimum temporal path cover (TPC) and temporally disjoint path cover (TDPC) in temporal graphs. Under the Dilworth property, it establishes for the first time that the size of a minimum cover equals the Lovász number of the underlying connected graph, thereby proving that both TPC and TDPC are solvable in polynomial time under this condition. Through parameterized complexity analysis, the work further shows that TPC becomes W[1]-hard when parameterized by the deletion distance to a linear forest, even with only two time steps, yet admits a fixed-parameter tractable (FPT) algorithm with respect to the vertex cover number. These results demonstrate that disregarding the number of time steps renders the problem NP-hard, highlighting the critical role of temporal structure in determining computational complexity.

Abstract

The Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) problems were introduced by [Chakraborty, Dailly, Foucaud, Klasing, MFCS '24]. Both were shown to be NP-hard on temporal DAGs, while the latter is also NP-hard on temporal oriented trees. All tractable cases for T(D)PC established in that paper satisfy a temporal Dilworth property, namely that the size of the minimum T(D)PC is equal to the size of the maximum antichain. This raises a natural question: is T(D)PC polynomial-time solvable under the promise that the respective Dilworth property holds? In this work, we answer this question in the affirmative for both problems, proving in fact that, under the respective promise, the size of the minimum T(D)PC is exactly equal to the Lovász number of the connectivity graph. 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.

Problem

• Temporal Path Cover
• Dilworth Property
• Parameterized Complexity
• NP-hardness
• Lovász Number

Innovation

• Temporal Path Cover
• Dilworth property
• Lovász number
• parameterized complexity
• W[1]-hardness