Résumé
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.