We study the complexity-theoretic boundaries of tractability for three classical problems in the context of Hierarchical Task Network Planning: the validation of a provided plan, whether an executable plan exists, and whether a given state can be reached by some plan. We show that all three problems can be solved in polynomial time on primitive task networks of constant partial order width (and a generalization thereof), whereas for the latter two problems this holds only under a provably necessary restriction to the state space. Next, we obtain an algorithmic meta-theorem along with corresponding lower bounds to identify tight conditions under which general polynomial-time solvability results can be lifted from primitive to general task networks. Finally, we enrich our investigation by analyzing the parameterized complexity of the three considered problems, and show that (1) fixed-parameter tractability for all three problems can be achieved by replacing the partial order width with the vertex cover number of the network as the parameter, and (2) other classical graph-theoretic parameters of the network (including treewidth, treedepth, and the aforementioned partial order width) do not yield fixed-parameter tractability for any of the three problems.

翻译：我们研究了层次任务网络规划背景下三个经典问题的计算复杂性可解性边界：给定计划的验证、可执行计划的存在性以及给定状态是否可以通过某个计划到达。我们证明，在原始任务网络具有常数偏序宽度（及其推广形式）时，这三个问题均可在多项式时间内解决；而对于后两个问题，该结论仅在状态空间满足可证明的必要限制下成立。进一步地，我们提出算法元定理并给出相应下界，以刻画将多项式时间可解性结果从原始任务网络推广到一般任务网络的严格条件。最后，我们通过分析三个问题的参数化复杂性丰富了研究，结果表明：(1) 以网络的顶点覆盖数替代偏序宽度作为参数时，三个问题均可实现固定参数可解性；(2) 网络的其他经典图论参数（包括树宽、树深度以及前述偏序宽度）无法为这三个问题中的任何一个提供固定参数可解性。