The class XNLP consists of (parameterized) problems that can be solved nondeterministically in $f(k)n^{O(1)}$ time and $f(k)\log n$ space, where $n$ is the size of the input instance and $k$ the parameter. The class XALP consists of problems that can be solved in the above time and space with access to an additional stack. These two classes are a "natural home" for many standard graph problems and their generalizations. In this paper, we show the hardness of several problems on planar graphs, parameterized by outerplanarity, treewidth and pathwidth, thus strengthening several existing results. In particular, we show the XNLP-hardness of the following problems parameterized by outerplanarity: All-or-Nothing Flow, Target Outdegree Orientation, Capacitated (Red-Blue) Dominating Set, Target Set Selections etc. We also show the XNLP-completeness of Scattered Set parameterized by pathwidth and XALP-completeness parameterized by treewidth and outerplanarity.

翻译：XNLP类包含那些可以在$f(k)n^{O(1)}$时间和$f(k)\log n$空间内非确定性求解的（参数化）问题，其中$n$是输入实例的大小，$k$是参数。XALP类包含在相同时间和空间限制下、可额外访问一个栈的问题。这两个类是许多标准图问题及其推广的“自然归宿”。本文证明了平面图上若干问题关于外平面性、树宽和路径宽的难度，从而强化了多项现有结果。特别地，我们证明了以下参数化问题关于外平面性的XNLP难度：全或无流（All-or-Nothing Flow）、目标出度定向（Target Outdegree Orientation）、容量化（红蓝）支配集（Capacitated (Red-Blue) Dominating Set）、目标集选择（Target Set Selections）等。我们还证明了分散集（Scattered Set）关于路径宽的XNLP完全性，以及关于树宽和外平面性的XALP完全性。