Computing Hamiltonian Paths with Partial Order Restrictions

TL;DR

This paper investigates the complexity of finding Hamiltonian paths under partial order constraints, proving NP-completeness in certain cases, and providing efficient algorithms for specific graph classes like outerplanar graphs.

Contribution

It establishes NP-completeness for Hamiltonian paths with partial order restrictions on bipartite graphs and posets of height 2, and offers an efficient algorithm for outerplanar graphs.

Findings

NP-completeness for bipartite graphs and height-2 posets

No significantly faster algorithm for width-k posets under ETH

Quadratic time algorithm for outerplanar graphs

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.