Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Bergougnoux, Benjamin; Chekan, Vera; Ganian, Robert; Kanté, Mamadou Moustapha; Mnich, Matthias; Oum, Sang-il; Pilipczuk, Michał; van Leeuwen, Erik Jan
(2023)
31st Annual European Symposium on Algorithms, ESA 2023, volume 274, pp. 18:1 - 18:18
Leibniz International Proceedings in Informatics, LIPIcs, volume 274, pp. 18:1 - 18:18
(Part of book)
Abstract
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that
... read more
this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, Independent Set can be solved in time 2O(dk) · nO(1) using O(dk2 log n) space; Max Cut can be solved in time nO(dk) using O(dk log n) space; and Dominating Set can be solved in time 2O(dk) · nO(1) using nO(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial.
show less
Download/Full Text
Keywords: Parameterized complexity, shrubdepth, space complexity, algebraic methods
ISSN: 1868-8969
ISBN: 978-3-95977-295-2
9783959772952
Publisher: Schloss Dagstuhl -- Leibniz-Zentrum für Informatik
Note: Funding Information: Funding Vera Chekan: Supported by the DFG Research Training Group 2434 “Facets of Complexity”. Robert Ganian: Project No. Y1329 of the Austrian Science Fund (FWF), WWTF Project ICT22-029. Mamadou Moustapha Kanté: Supported by the French National Research Agency (ANR-18-CE40-0025-01 and ANR-20-CE48-0002). Sang-il Oum: Supported by the Institute for Basic Science (IBS-R029-C1). Michał Pilipczuk: This work is a part of project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057). Funding Information: Vera Chekan: Supported by the DFG Research Training Group 2434 “Facets of Complexity”. Robert Ganian: Project No. Y1329 of the Austrian Science Fund (FWF), WWTF Project ICT22-029. Mamadou Moustapha Kanté: Supported by the French National Research Agency (ANR-18-CE40-0025-01 and ANR-20-CE48-0002). Sang-il Oum: Supported by the Institute for Basic Science (IBS-R029-C1). Michał Pilipczuk: This work is a part of project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation pro-gramme (grant agreement No. 948057). Publisher Copyright: © Benjamin Bergougnoux, Vera Chekan, Robert Ganian, Mamadou Moustapha Kanté, Matthias Mnich, Sang-il Oum, Michał Pilipczuk, and Erik Jan van Leeuwen.
(Peer reviewed)
See more statistics about this item