Algorithms and complexity for path covers of temporal DAGs: when is Dilworth dynamic?

TL;DR

This work investigates two dynamic path-cover problems on temporal DAGs: Temporal Path Cover ($TPC$) and Temporally Disjoint Path Cover ($TD-PC$). It combines hardness results (NP-hardness even under strong restrictions and for temporal oriented trees) with structural and algorithmic strategies, including a reduction to clique covers on weakly chordal graphs to obtain a polynomial-time $TPC$ algorithm for temporal oriented trees, and a treewidth/time-based dynamic programming framework to achieve FPT/XP results for general temporal digraphs. The paper also delineates tractable subclasses (temporal oriented lines and temporal rooted directed trees) that satisfy the TD-Dilworth property, and establishes an explicit DP-based approach on tree decompositions for bounded-treewidth cases. Overall, it reveals a nuanced landscape where Dilworth-type results enable efficient covers in certain temporal tree-like structures, while the temporally disjoint version remains hard in many natural classes, with parameterized tractability tied to structural graph parameters like treewidth and time-labels.

Abstract

In this paper, we study a dynamic analogue of the Path Cover problem, which can be solved in polynomial-time in directed acyclic graphs. A temporal digraph has an arc set that changes over discrete time-steps, if the underlying digraph (the union of all the arc sets) is acyclic, then we have a temporal DAG. A temporal path is a directed path in the underlying digraph, such that the time-steps of arcs are strictly increasing along the path. Two temporal paths are temporally disjoint if they do not occupy any vertex at the same time. A temporal (resp. temporally disjoint) path cover is a collection of (resp. temporally disjoint) temporal paths that covers all vertices. In this paper, we study the computational complexities of the problems of finding a temporal (disjoint) path cover with minimum cardinality, denoted as Temporal Path Cover (TPC) and Temporally Disjoint Path Cover (TD-PC). We show that both problems are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, TD-PC remains NP-hard even on temporal oriented trees. In contrast, we show that TPC is polynomial-time solvable on temporal oriented trees by a reduction to Clique Cover for (static undirected) weakly chordal graphs (a subclass of perfect graphs for which Clique Cover admits an efficient algorithm). This highlights an interesting algorithmic difference between the two problems. Although it is NP-hard on temporal oriented trees, TD-PC becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees. We also show that TPC (resp. TD-PC) admits an XP (resp. FPT) time algorithm with respect to parameter tmax + tw, where tmax is the maximum time-step, and tw is the treewidth of the underlying static undirected graph.

Figures (7)

• Figure 1: A minimum-size (temporally disjoint) temporal path cover is shown, vertices in a maximum-size temporal antichain are in black.
• Figure 2: The gadget $H(s_i)$.
• Figure 3: Vertex partition of the gadget $H(s_i)$ into length-2 paths.
• Figure 4: $\mathcal{T}_n$, a temporal DAG with a maximum-size temporal antichain of size 1 and a minimum-size temporal path cover of size $\left\lceil \frac{n}{2} \right\rceil$, for $n=4,5$.
• Figure 5: Layer 1 of the reduction of the proof of \ref{['thm-orientedTreesNP']}, with $x_1=3$, $b=3$, $B=4$. The only $v_j^k$'s and $w_j^k$'s linked with $c$ in this layer are those with $j=1$.

Theorems & Definitions (43)

• Theorem 1: Dilworth dilworth
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 7
• Theorem 8: HaywardSS07lovasz1972normalSpinradS95
• Theorem 8
• proof