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FPT Approximation of Generalised Hypertree Width for Bounded Intersection Hypergraphs

Authors Matthias Lanzinger , Igor Razgon

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LIPIcs.STACS.2024.48.pdf
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Author Details

Matthias Lanzinger
  • TU Wien, Austria
  • University of Oxford, UK
Igor Razgon
  • Birkbeck, University of London, UK

Funding

  • Lanzinger, Matthias: Matthias Lanzinger acknowledges support by the Royal Society "RAISON DATA" project (Reference No. RPbackslashR1backslash201074). Lanzinger has been supported by the Vienna Science and Technology Fund (WWTF) [10.47379/ICT2201].

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Matthias Lanzinger and Igor Razgon. FPT Approximation of Generalised Hypertree Width for Bounded Intersection Hypergraphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.48

Abstract

Generalised hypertree width (ghw) is a hypergraph parameter that is central to the tractability of many prominent problems with natural hypergraph structure. Computing ghw of a hypergraph is notoriously hard. The decision version of the problem, checking whether ghw(H) ≤ k, is paraNP-hard when parameterised by k. Furthermore, approximation of ghw is at least as hard as approximation of Set-Cover, which is known to not admit any FPT approximation algorithms. Research in the computation of ghw so far has focused on identifying structural restrictions to hypergraphs - such as bounds on the size of edge intersections - that permit XP algorithms for ghw. Yet, even under these restrictions that problem has so far evaded any kind of FPT algorithm. In this paper we make the first step towards FPT algorithms for ghw by showing that the parameter can be approximated in FPT time for graphs of bounded edge intersection size. In concrete terms we show that there exists an FPT algorithm, parameterised by k and d, that for input hypergraph H with maximal cardinality of edge intersections d and integer k either outputs a tree decomposition with ghw(H) ≤ 4k(k+d+1)(2k-1), or rejects, in which case it is guaranteed that ghw(H) > k. Thus, in the special case of hypergraphs of bounded edge intersection, we obtain an FPT O(k³)-approximation algorithm for ghw.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Hypergraphs
Keywords
  • generalized hypertree width
  • hypergraphs
  • parameterized algorithms
  • approximation algorithms

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References

  1. Isolde Adler, Georg Gottlob, and Martin Grohe. Hypertree width and related hypergraph invariants. Eur. J. Comb., 28(8):2167-2181, 2007. URL: https://doi.org/10.1016/j.ejc.2007.04.013.
  2. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  3. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  4. Grzegorz Fabianski, Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. Progressive algorithms for domination and independence. In Rolf Niedermeier and Christophe Paul, editors, STACS 2019, volume 126 of LIPIcs, pages 27:1-27:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.STACS.2019.27.
  5. Wolfgang Fischl, Georg Gottlob, Davide Mario Longo, and Reinhard Pichler. Hyperbench: A benchmark and tool for hypergraphs and empirical findings. ACM J. Exp. Algorithmics, 26:1.6:1-1.6:40, 2021. URL: https://doi.org/10.1145/3440015.
  6. Wolfgang Fischl, Georg Gottlob, and Reinhard Pichler. General and fractional hypertree decompositions: Hard and easy cases. In Proceedings PODS, pages 17-32. ACM, 2018. URL: https://doi.org/10.1145/3196959.3196962.
  7. Georg Gottlob and Gianluigi Greco. Decomposing combinatorial auctions and set packing problems. J. ACM, 60(4):24:1-24:39, 2013. URL: https://doi.org/10.1145/2508028.2505987.
  8. Georg Gottlob, Gianluigi Greco, and Francesco Scarcello. Pure nash equilibria: Hard and easy games. J. Artif. Intell. Res., 24:357-406, 2005. URL: https://doi.org/10.1613/jair.1683.
  9. Georg Gottlob, Martin Grohe, Nysret Musliu, Marko Samer, and Francesco Scarcello. Hypertree decompositions: Structure, algorithms, and applications. In WG, volume 3787 of Lecture Notes in Computer Science, pages 1-15. Springer, 2005. URL: https://doi.org/10.1007/11604686_1.
  10. Georg Gottlob, Matthias Lanzinger, Cem Okulmus, and Reinhard Pichler. Fast parallel hypertree decompositions in logarithmic recursion depth. In Proceedings PODS, pages 325-336. ACM, 2022. URL: https://doi.org/10.1145/3517804.3524153.
  11. Georg Gottlob, Matthias Lanzinger, Reinhard Pichler, and Igor Razgon. Complexity analysis of generalized and fractional hypertree decompositions. J. ACM, 68(5):38:1-38:50, 2021. URL: https://doi.org/10.1145/3457374.
  12. Georg Gottlob, Nicola Leone, and Francesco Scarcello. Robbers, marshals, and guards: game theoretic and logical characterizations of hypertree width. J. Comput. Syst. Sci., 66(4):775-808, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00030-8.
  13. Georg Gottlob, Zoltán Miklós, and Thomas Schwentick. Generalized hypertree decompositions: NP-hardness and tractable variants. J. ACM, 56(6):30:1-30:32, 2009. URL: https://doi.org/10.1145/1568318.1568320.
  14. Georg Gottlob, Cem Okulmus, and Reinhard Pichler. Fast and parallel decomposition of constraint satisfaction problems. Constraints An Int. J., 27(3):284-326, 2022. URL: https://doi.org/10.1007/s10601-022-09332-1.
  15. Martin Grohe and Dániel Marx. Constraint solving via fractional edge covers. ACM Trans. Algorithms, 11(1):4:1-4:20, 2014. URL: https://doi.org/10.1145/2636918.
  16. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1-33:38, 2019. URL: https://doi.org/10.1145/3325116.
  17. Matthias Lanzinger and Igor Razgon. FPT approximation of generalised hypertree width for bounded intersection hypergraphs. CoRR, abs/2309.17049, 2023. URL: https://doi.org/10.48550/ARXIV.2309.17049.
  18. Dániel Marx. Approximating fractional hypertree width. ACM Trans. Algorithms, 6(2):29:1-29:17, 2010. URL: https://doi.org/10.1145/1721837.1721845.
  19. Dan Olteanu and Maximilian Schleich. Factorized databases. SIGMOD Rec., 45(2):5-16, 2016. URL: https://doi.org/10.1145/3003665.3003667.
  20. Geevarghese Philip, Venkatesh Raman, and Somnath Sikdar. Solving dominating set in larger classes of graphs: FPT algorithms and polynomial kernels. In Proceedings ESA, volume 5757 of Lecture Notes in Computer Science, pages 694-705. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-04128-0_62.
  21. Igor Razgon. FPT algoritms providing constant ratio approximation of hypertree width parameters for hypergraphs of bounded rank. CoRR, abs/2212.13423, 2022. URL: https://doi.org/10.48550/arXiv.2212.13423.
  22. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
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