On the Complexity of Problems on Tree-structured Graphs

TL;DR

The paper introduces XALP, a parameterized complexity class defined as NAuxPDA with time $f(k)n^{O(1)}$ and space $f(k) obreak ext{log} obreak n$, capturing tree-structured problems. It provides multiple equivalent characterisations (including ATM-based tree-size and stack-based simulations) and proves XALP-completeness for natural problems on tree-structured graphs, such as List Colouring and All-or-Nothing Flow parameterized by treewidth, Independent Set and Dominating Set parameterized by treewidth/$ obreak ext{log} obreak n$, and Max Cut parameterized by cliquewidth. The paper also introduces tree-shaped variants of Weighted CNF-Satisfiability and Multicolour Clique, and demonstrates a broad set of XALP-hardness results (e.g., Tree-Chained Multicolour Clique/Independent Set, Binary CSP) to establish XALP as the natural home for many tree-structured parameterized problems. These results offer a structured framework for reductions and provide insight into the limitations of XP-space algorithms for such problems, guiding future classification and algorithmic study in this domain.

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.

Figures (1)

• Figure 1: Local structure of a satisfying assignment in the constructed instance of Tree-Chained Multicolor Clique ($k'=4$). The blue edges enforce that the positions of the heads on tapes and the state of the TM are consistent. The black edges enforce that what is written on the work tape does not change in blocks where the head is not present. The red edges enforce that the state, head positions, and the bit at the position of the head on the work tape can be changed exactly by the transitions of the TM. We then add further edges to form cliques, but they do not enforce any constraints.

Theorems & Definitions (14)

• Theorem 1
• Lemma 2
• Claim 4
• Claim 5
• Theorem 6
• Corollary 7
• Corollary 8
• Theorem 9
• Theorem 10
• Theorem 12