A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles I: Treewidth, Pathwidth, and Grid Graphs

TL;DR

This paper investigates the computational complexity of finding Hamiltonian paths and cycles with precedence constraints in graphs, revealing NP-completeness in certain graph classes and providing polynomial algorithms for others, with implications for grid graphs.

Contribution

It establishes new NP-completeness results and polynomial-time algorithms for Hamiltonian problems under partial order constraints based on graph width parameters.

Findings

NP-complete for pathwidth 4 (paths) and 5 (cycles)

Polynomial algorithms for pathwidth 3 and treewidth 2 (paths), pathwidth 4 and treewidth 3 (cycles)

NP-complete on grid graphs with height ≥7 (paths) and ≥9 (cycles)

Abstract

We consider the problem of finding a Hamiltonian path or a Hamiltonian cycle with precedence constraints in the form of a partial order on the vertex set. We show that the path problem is $\mathsf{NP}$-complete for graphs of pathwidth 4 while the cycle problem is $\mathsf{NP}$-complete on graphs of pathwidth 5. We complement these results by giving polynomial-time algorithms for graphs of pathwidth 3 and treewidth 2 for Hamiltonian paths as well as pathwidth 4 and treewidth 3 for Hamiltonian cycles. Furthermore, we study the complexity of the path and cycle problems on rectangular grid graphs of bounded height. For these, we show that the path and cycle problems are $\mathsf{NP}$-complete when the height of the grid is greater or equal to 7 and 9, respectively. In the variant where we look for minimum edge-weighted Hamiltonian paths and cycles, the problems are $\mathsf{NP}$-hard for heights 5 and 6, respectively.