On space efficiency of algorithms working on structural decompositions of graphs

TL;DR

This paper explores the space complexity of algorithms on graph decompositions, linking it to longstanding conjectures and establishing a hierarchy of complexity classes based on decomposition parameters.

Contribution

It demonstrates the relationship between space efficiency in graph algorithms and a key conjecture, extending the complexity landscape to tree-depth decompositions.

Findings

Space complexity explosion is related to a conjecture on LCS problem.

Computations on tree-depth decompositions correspond to a specific nondeterministic machine model.

A hierarchy of complexity classes for graph parameters is established.

Abstract

Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition's width. Following the work of Allender et al. [Theory of Computing, '14], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we complete the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition's depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.