On the Complexity of Problems on Tree-structured Graphs

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 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.