On the Complexity of Problems on Tree-structured Graphs

Hans L. Bodlaender; Carla Groenland; Hugo Jacob; Marcin Pilipczuk; Michał Pilipczuk

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arXiv · 2206.11828

On the Complexity of Problems on Tree-structured Graphs

Hans L. Bodlaender·Carla Groenland·Hugo Jacob·Marcin Pilipczuk·Michał Pilipczuk

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Abstract

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.

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BibTeX RIS

Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Marcin Pilipczuk, Michał Pilipczuk. 2022-06-23. On the Complexity of Problems on Tree-structured Graphs. https://arxiv.org/abs/2206.11828

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