Saving Space by Dynamic Algebraization Based on Tree Decomposition: Minimum Dominating Set

TL;DR

This paper introduces a polynomial space algorithm for the Minimum Dominating Set problem using dynamic programming on tree decompositions, balancing space efficiency with moderate time complexity, especially effective for graphs with small tree depth.

Contribution

It presents a novel polynomial space algorithm for Minimum Dominating Set based on dynamic algebraization and tree decomposition, improving space efficiency over previous methods.

Findings

Runs in O*(3^d) time, where d is tree-depth.

Uses polynomial space O(nk), with k as treewidth.

Effective for graphs with small tree depth.

Abstract

An algorithm is presented that solves the Minimum Dominating Set problem exactly using polynomial space based on dynamic programming for a tree decomposition. A direct application of dynamic programming based on a tree decomposition would result in an exponential space algorithm, but we use zeta transforms to obtain a polynomial space algorithm in exchange for a moderate increase of the time. This framework was pioneered by Lokshtanov and Nederlof 2010 and adapted to a dynamic setting by F\"urer and Yu 2017. Our space-efficient algorithm is a parametrized algorithm based on tree-depth and treewidth. The naive algorithm for Minimum Dominating Set runs in $\mathcal{O}^*(2^n)$ time. Most of the previous works have focused on time complexity. But space optimization is a crucial aspect of algorithm design, since in several scenarios space is a more valuable resource than time. Our parametrized algorithm runs in $\mathcal{O}^*(3^{d})$, and its space complexity is $\mathcal{O}(nk)$, where $d$ is the depth and $k$ is the width of the given tree decomposition. We observe that Reed's 1992 algorithm constructing a tree decomposition of a graph uses only polynomial space. So, even if the tree decomposition is not given, we still obtain an efficient polynomial space algorithm. There are some other algorithms which use polynomial space for this problem, but they are not efficient for graphs with small tree depth.