Space-Efficient Vertex Separators for Treewidth

For n -vertex graphs with treewidth k = O ( n 1 / 2 - ϵ ) and an arbitrary ϵ > 0 , we present a word-RAM algorithm to compute vertex separators using only O ( n ) bits of working memory. As an application of our algorithm, we give an O (1)-approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in c k n ( log log n ) log ∗ n time using O ( n ) bits for some constant c > 0 . Together with the result of Banerjee et al. (Proceedings of 21st international conference on computing and combinatorics (COCOON 2015). LNCS, vol 9198, Springer, pp 349–360, 2015. https://doi.org/10.1007/978-3-319-21398-9_28 ) we are able to compute a solution for all monadic-second-order problems (MSO) with O ( n + τ ( k ) · p ( log p n ) log n ) bits in O ( τ ( k ) · n 2 + ( 2 / log p ) ) time where k is the treewidth of the given graph, p is some arbitrary parameter with 2 ≤ p ≤ n and τ is some function depending on the MSO formula. We finally use the tree decomposition obtained by our algorithm to solve Vertex Cover , Independent Set , Dominating Set , MaxCut and q - Coloring by using polynomial time and O ( n ) bits as long as the treewidth of the graph is smaller than c ′ log n for some problem dependent constant 0 < c ′ < 1 .