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As an application of our algorithm, we give an O<\/jats:italic>(1)-approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in $$c^k n (\\log \\log n) \\log ^* n$$<\/jats:tex-math>\n \n \n c<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msup>\n n<\/mml:mi>\n \n (<\/mml:mo>\n log<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n log<\/mml:mo>\n \u2217<\/mml:mo>\n <\/mml:msup>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> time using O<\/jats:italic>(n<\/jats:italic>) bits for some constant $$c > 0$$<\/jats:tex-math>\n \n c<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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\u2013360, 2015. https:\/\/doi.org\/10.1007\/978-3-319-21398-9_28<\/jats:ext-link>) we are able to compute a solution for all monadic-second-order problems (MSO) with $$O(n + \\tau (k) \\cdot p (\\log _{p} n) \\log n)$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n \u03c4<\/mml:mi>\n \n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u00b7<\/mml:mo>\n p<\/mml:mi>\n \n (<\/mml:mo>\n \n log<\/mml:mo>\n p<\/mml:mi>\n <\/mml:msub>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> bits in $$O(\\tau (k) \\cdot n^{2 + (2\/\\log p)})$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \u03c4<\/mml:mi>\n \n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u00b7<\/mml:mo>\n \n n<\/mml:mi>\n \n 2<\/mml:mn>\n +<\/mml:mo>\n (<\/mml:mo>\n 2<\/mml:mn>\n \/<\/mml:mo>\n log<\/mml:mo>\n p<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> time where k<\/jats:italic> is the treewidth of the given graph, p<\/jats:italic> is some arbitrary parameter with $$2 \\le p \\le n$$<\/jats:tex-math>\n \n 2<\/mml:mn>\n \u2264<\/mml:mo>\n p<\/mml:mi>\n \u2264<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\tau $$<\/jats:tex-math>\n \u03c4<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is some function depending on the MSO formula. We finally use the tree decomposition obtained by our algorithm to solve Vertex Cover<\/jats:sc>, Independent Set<\/jats:sc>, Dominating Set<\/jats:sc>, MaxCut<\/jats:sc> and q<\/jats:italic>-Coloring<\/jats:sc> by using polynomial time and O<\/jats:italic>(n<\/jats:italic>) bits as long as the treewidth of the graph is smaller than $$c' \\log n$$<\/jats:tex-math>\n \n \n c<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for some problem dependent constant $$0< c' < 1$$<\/jats:tex-math>\n \n 0<\/mml:mn>\n <<\/mml:mo>\n \n c<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00453-022-00967-3","type":"journal-article","created":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T08:04:31Z","timestamp":1651219471000},"page":"2414-2461","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Space-Efficient Vertex Separators for Treewidth"],"prefix":"10.1007","volume":"84","author":[{"given":"Frank","family":"Kammer","sequence":"first","affiliation":[]},{"given":"Johannes","family":"Meintrup","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5946-8087","authenticated-orcid":false,"given":"Andrej","family":"Sajenko","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,29]]},"reference":[{"key":"967_CR1","volume-title":"Network Flows: Theory, Algorithms and Applications","author":"RK Ahuja","year":"1993","unstructured":"Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. 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