On the Complexity of Problems on Tree-structured Graphs

TL;DR

This paper introduces the class XALP, characterizes its problems, and shows many natural problems on tree-structured graphs are complete for this class, expanding understanding of parameterized complexity with tree-structured constraints.

Contribution

The paper defines the new class XALP, provides multiple characterizations, and proves several natural problems are XALP-complete, establishing a framework for future reductions in tree-structured graph problems.

Findings

XALP is characterized as problems solvable by an alternating Turing machine with bounded tree size and logarithmic space.

Natural problems like List Colouring and All-or-Nothing Flow are XALP-complete when parameterized by treewidth.

Tree-shaped variants of Weighted CNF-Satisfiability and Multicolour Clique are also XALP-complete.

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.