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Dynamic Programming on Bipartite Tree Decompositions

Authors Lars Jaffke, Laure Morelle, Ignasi Sau, Dimitrios M. Thilikos

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Author Details

Lars Jaffke
  • Department of Informatics, University of Bergen, Norway
Laure Morelle
  • LIRMM, Université de Montpellier, CNRS, France
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France
Dimitrios M. Thilikos
  • LIRMM, Université de Montpellier, CNRS, France

Funding

The second and the third authors were supported by the ANR project ELIT (ANR-20-CE48-0008-01), the three last authors were supported by the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027), and the first author was supported by the Research Council of Norway (No 274526).

Acknowledgements

We thank Sebastian Wiederrecht and the reviewers for helpful remarks.

Cite AsGet BibTex

Lars Jaffke, Laure Morelle, Ignasi Sau, and Dimitrios M. Thilikos. Dynamic Programming on Bipartite Tree Decompositions. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 26:1-26:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.26

Abstract

We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Theor. Comput. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that K_t-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are FPT parameterized by bipartite treewidth. We also provide the following complexity dichotomy when H is a 2-connected graph, for each of the H-Subgraph-Packing, H-Induced-Packing, H-Scattered-Packing, and H-Odd-Minor-Packing problems: if H is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if H is non-bipartite, then the problem is solvable in XP-time. Beyond bipartite treewidth, we define 1-ℋ-treewidth by replacing the bipartite graph class by any graph class ℋ. Most of the technology developed here also works for this more general parameter.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • tree decomposition
  • bipartite graphs
  • dynamic programming
  • odd-minors
  • packing
  • maximum cut
  • vertex cover
  • independent set
  • odd cycle transversal

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