A Graph Width Perspective on Partially Ordered Hamiltonian Paths

TL;DR

This work systematically maps the computational landscape of the Partially Ordered Hamiltonian Path Problem (POHPP) across graph width and distance parameters. It proves NP-hardness on width-bounded graphs at and above certain thresholds (e.g., pathwidth 4, bandwidth-related cases) while delivering efficient algorithms for key tractable regimes (pathwidth 3, treewidth 2, treedepth-bound cases). It further distinguishes the complexity under various distance-to-class measures, showing $ extsf{W[1]}$-hardness for some distances and providing FPT/XP algorithms for others (notably, feedback edge set and block-distance regimes). The paper also introduces DP- and module-based techniques, including signature-based encodings and a method for integrating clique structure into dynamic programs, to achieve fixed-parameter tractability or polynomial-time results in several non-sparse settings. Collectively, the results illuminate where POHPP remains intractable and where efficient, parameterized algorithms can be achieved, guiding future work on treewidth, grid graphs, and planar/outerplanar settings.

Abstract

We consider the problem of finding a Hamiltonian path with precedence constraints in the form of a partial order on the vertex set. This problem is known as Partially Ordered Hamiltonian Path Problem (POHPP). Here, we study the complexity for graph width parameters for which the ordinary Hamiltonian Path problem is in $\mathsf{FPT}$. We show that POHPP is $\mathsf{NP}$-complete for graphs of pathwidth 4. We complement this result by giving polynomial-time algorithms for graphs of pathwidth 3 and treewidth 2. Furthermore, we show that POHPP is $\mathsf{NP}$-hard for graphs of clique cover number 2 and $\mathsf{W[1]}$-hard for some distance-to-$\mathcal{G}$ parameters, including distance to path and distance to clique. In addition, we present $\mathsf{XP}$ and $\mathsf{FPT}$ algorithms for parameters such as distance to block and feedback edge set number.

Figures (12)

• Figure 1: Diagram illustrating the complexity results for (Min)POHPP for different graph width parameters. A directed solid edge from parameter $P$ to parameter $Q$ means that a bounded value of $P$ implies a bounded value for $Q$. A directed dashed edge implies that this relation does not hold in general but for traceable graphs, i.e., graphs having a Hamiltonian path. If a directed solid path from $P$ to $Q$ is missing, then parameter $Q$ is unbounded for the graphs of bounded $P$. The same holds for the traceable graphs if there is also no path using dashed edges.
• Figure 2: Complete construction of the proof of \ref{['thm:np-interval']}. The boxes mark the clause gadgets.
• Figure 3: The gadgets for \ref{['thm:np-grid']} combined to represent a formula. The gray vertices come after $t$ in the partial order $\pi$. A black square is negative variable vertex, a white square is a positive variable vertex. A white disk marks a literal vertex.
• Figure 4: The path $\mathcal{P}_{s,t}$ is stuck when traversing both row $3$ and row $5$ of $X_i$
• Figure 5: The path $\mathcal{P}$ in the start, middle and end gadget

Theorems & Definitions (44)

• theorem 1
• proof
• theorem 2
• proof
• theorem 3
• proof
• proof
• lemma 1
• proof
• lemma 2