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 Colouring 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 Multicolour Clique that are XALP-complete.

翻译：本文引入了一类新的参数化问题，称之为XALP：所有可由非确定型图灵机在辅助栈（仅允许查看栈顶元素）支持下，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。多种关于“树状结构图”的自然问题对该类具有完备性：我们证明以树宽为参数的列表染色问题和全有或全无流问题为XALP-完备。此外，以树宽除以$\log n$为参数的独立集问题和支配集问题，以及以团宽为参数的最大割问题同样为XALP-完备。除为这些问题找到“自然归宿”外，我们亦为后续归约铺平道路。我们给出了XALP类的若干等价刻画，例如XALP即为可由交替型图灵机求解且其运行过程的树规模不超过$f(k)n^{O(1)}$、空间使用不超过$f(k)\log n$的问题类。同时，我们引入了加权可满足性问题和多色团问题的“树状”变体，并证明其为XALP-完备。