Space-Efficient Vertex Separators for Treewidth

Abstract

For n-vertex graphs with treewidth k = O(n^{1/2-epsilon }) and an arbitrary epsilon >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 + tau (k) cdot p (log _{p} n) log n) bits in O(tau (k) cdot n^{2 + (2/log p)}) time where k is the treewidth of the given graph, p is some arbitrary parameter with 2 le p le n and tau 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.