Space-Efficient Vertex Separators for Treewidth

Abstract

AbstractFor n-vertex graphs with treewidth $$k = O(n^{1/2-\epsilon })$$ k = O ( n 1 / 2 - ϵ ) and an arbitrary $$\epsilon >0$$ 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 $$c^k n (\log \log n) \log ^* n$$ c k n ( log log n ) log ∗ n time using O(n) bits for some constant $$c > 0$$ 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 $$O(n + \tau (k) \cdot p (\log _{p} n) \log n)$$ O ( n + τ ( k ) · p ( log p n ) log n ) bits in $$O(\tau (k) \cdot n^{2 + (2/\log p)})$$ O ( τ ( k ) · n 2 + ( 2 / log p ) ) time where k is the treewidth of the given graph, p is some arbitrary parameter with $$2 \le p \le n$$ 2 ≤ p ≤ n and $$\tau $$ 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 $$c' \log n$$ c ′ log n for some problem dependent constant $$0< c' < 1$$ 0 < c ′ < 1 .