🤖 AI Summary
This paper investigates the Hamiltonian Path problem with partial order constraints: finding a path visiting all vertices of a given graph while respecting precedence constraints of the form “u must be visited before v.” Methodologically, it integrates combinatorial graph theory, partial order theory, parameterized algorithm design, and conditional lower-bound analysis under the Exponential Time Hypothesis (ETH). The main contributions are threefold: (1) It establishes NP-completeness even on complete bipartite graphs and partial orders of height 2, yielding tight complexity boundaries. (2) For partial orders of width k, it proves an Ω(nᵏ) time lower bound under ETH—optimal up to constant factors—and rules out algorithms asymptotically faster than O(k²nᵏ). (3) It presents the first O(n²) exact algorithm for outerplanar graphs, substantially expanding the class of efficiently solvable instances.
📝 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
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.