In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in $f(k)n^{O(1)}$ time and $f(k)\log n$ space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on `tree-structured graphs' are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by $\log n$, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a `natural home' for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most $f(k)n^{O(1)}$ and use $f(k)\log n$ space. Moreover, we introduce `tree-shaped' variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.

翻译：本文引入了一类新的参数化问题，称为XALP：所有可在非确定性图灵机上借助辅助栈（仅允许查看栈顶元素）以$f(k)n^{O(1)}$时间和$f(k)\log n$空间求解的参数化问题构成的类。许多关于“树结构图”的自然问题对此类具有完备性：我们证明了以树宽参数化的列表染色问题和全有或全无流问题是XALP完全的。此外，以树宽除以$\log n$参数化的独立集问题和支配集问题，以及以团宽参数化的最大割问题同样是XALP完全的。除为这些问题寻找“自然归宿”外，我们还为后续归约铺平了道路。我们给出了XALP类的若干等价刻画，例如XALP等价于可由交替式图灵机求解的问题类，该图灵机的运行树规模不超过$f(k)n^{O(1)}$且使用$f(k)\log n$空间。同时，我们引入了加权CNF可满足性问题和多色团问题的“树形”变体，并证明它们为XALP完全问题。