Computing Hamiltonian Paths with Partial Order Restrictions

TL;DR

This work studies the Partially Ordered Hamiltonian Path Problem ($POHPP$), asking whether a graph admits a Hamiltonian path whose vertex order extends a given partial order. It maps the tractability landscape: $POHPP$ remains $NP$-hard even on restricted graphs such as balanced complete bipartite graphs and height-$2$ posets, but becomes tractable when the poset width is bounded only to reveal an $O(k^2 n^k)$ XP algorithm (with ETH-based lower bounds ruling out significant improvements). It also identifies a fixed-parameter tractable approach based on the distance to linear order, and provides a quadratic-time algorithm for $POHPP$ on outerplanar graphs, thereby clarifying which graph- and poset-structure parameters yield efficient solutions. Together, these results delineate the boundaries between intractability and tractability for precedence-constrained Hamiltonian paths and point to fruitful directions for poset-dimension and graph-class research, including directed variants.

Abstract

When solving the Hamiltonian path problem it seems natural to be given additional precedence constraints for the order in which the vertices are visited. For example one could decide whether a Hamiltonian path exists for a fixed starting point, or that some vertices are visited before another vertex. We consider the problem of finding a Hamiltonian path that observes all precedence constraints given in a partial order on the vertex set. We show that this problem is $\mathsf{NP}$-complete even if restricted to complete bipartite graphs and posets of height 2. In contrast, for posets of width $k$ there is a known $\mathcal{O}(k^2 n^k)$ algorithm for arbitrary graphs with $n$ vertices. We show that it is unlikely that the running time of this algorithm can be improved significantly, i.e., there is no $f(k) n^{o(k)}$ time algorithm under the assumption of the Exponential Time Hypothesis. Furthermore, for the class of outerplanar graphs, we give an $\mathcal{O}(n^2)$ algorithm for arbitrary posets.

Figures (3)

• Figure 1: Overview of the graph $G'$. A box with rounded corners represents a clique, a box with square corners represents an independent set. An edge to a box symbolizes edges to all vertices contained in it. Note that the edges between the $W^i$ have been omitted for the sake of clarity. A visualization of these edges can be found in \ref{['width:fig2']}.
• Figure 2: On the left: Closeup of $G'$ for $k=4$ and the 4-clique $\{v^{1}_{p_{1}},v^{2}_{p_{2}},v^{3}_{p_{3}},v^{4}_{p_{4}}\}$. Each column $i$ represents the vertices $U^i_{p_i} \setminus \{w^i_{p_i,0}\} = \{w^i_{p_i,\ell}~|~\ell \in \{1,\ldots,4\}\}$. Note that the edge set of the induced subgraph of $G'$ belonging to this clique forms a path from $w^{1}_{p_1,1}$ to $w^{4}_{p_4,4}$. The blue segment of the path checks whether $v^1_{p_1}$ is adjacent to all other vertices, the green segment checks whether $v^2_{p_2}$ is adjacent to both $v^3_{p_3}$ and $v^4_{p_4}$, and the pink segment checks whether $v^3_{p_3}$ is adjacent to $v^4_{p_4}$. On the right: Here we denote the different types of edges. Blue edges are derived from Rule \ref{['e7']}. Green edges are derived from Rule \ref{['e8']}. Pink edges are derived from Rule \ref{['e9']}.
• Figure 3: One step in \ref{['algo:outer']}. The solid interval represents the interval $[a,b]$ for which we want to compute the entry in $M$. The left graph shows the case where $\omega$ is $1$ and the right graph shows the case where $\omega$ is $2$. The vertices that should be last in the subpath are filled gray. The dotted intervals represent the respective interval for which the entry of $M$ is checked. In both cases, we check whether the edge $ab$ exists.

Theorems & Definitions (39)

• theorem 1: Dilworth dilworth1987decomposition
• lemma 1
• proof
• lemma 2
• proof
• theorem 2
• theorem 3
• proof
• corollary 1
• theorem 4