  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.
