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# Computing Tree Decompositions with Small Independence Number

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## Cite As

Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Martin Milanič. Computing Tree Decompositions with Small Independence Number. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.51

## Abstract

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum weight independent set, can be solved in time n^𝒪(k) if the input n-vertex graph is given together with a tree decomposition of independence number k. Yolov in [SODA 2018] gave an algorithm that given an n-vertex graph G and an integer k, in time n^𝒪(k³) either constructs a tree decomposition of G whose independence number is 𝒪(k³) or correctly reports that the tree-independence number of G is larger than k. In this paper, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time 2^𝒪(k²) n^𝒪(k) and either outputs a tree decomposition of G with independence number at most 8k, or determines that the tree-independence number of G is larger than k. This implies 2^𝒪(k²) n^𝒪(k)-time algorithms for various problems, like maximum weight independent set, parameterized by the tree-independence number k without needing the decomposition as an input. Assuming Gap-ETH, an n^Ω(k) factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k ≥ 4 it is NP-hard to decide if a given graph has the tree-independence number at most k.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
• Theory of computation → Graph algorithms analysis
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• tree-independence number
• approximation
• parameterized algorithms

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## References

1. Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl, and Kristina Vušković. Tree independence number i. (even hole, diamond, pyramid)-free graphs, 2024. URL: https://arxiv.org/abs/2305.16258.
2. Isolde Adler. Width functions for hypertree decompositions. PhD thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2006.
3. Brenda S. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach., 41(1):153-180, 1994.
4. M. Bíró, M. Hujter, and Zs. Tuza. Precoloring extension. I. Interval graphs. Discrete Math., 100(1-3):267-279, 1992. URL: https://doi.org/10.1016/0012-365X(92)90646-W.
5. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM J. Comput., 45(2):317-378, 2016. URL: https://doi.org/10.1137/130947374.
6. Hans L. Bodlaender, Jens Gustedt, and Jan Arne Telle. Linear-time register allocation for a fixed number of registers. In Howard J. Karloff, editor, Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 25-27 January 1998, San Francisco, California, USA, pages 574-583. ACM/SIAM, 1998. URL: http://dl.acm.org/citation.cfm?id=314613.314994.
7. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more. SIAM J. Comput., 49(4):772-810, 2020. URL: https://doi.org/10.1137/18M1166869.
8. Steven Chaplick, Fedor V. Fomin, Petr A. Golovach, Dušan Knop, and Peter Zeman. Kernelization of graph Hamiltonicity: proper H-graphs. SIAM J. Discrete Math., 35(2):840-892, 2021. URL: https://doi.org/10.1137/19M1299001.
9. Steven Chaplick, Martin Töpfer, Jan Voborník, and Peter Zeman. On H-topological intersection graphs. Algorithmica, 83(11):3281-3318, 2021. URL: https://doi.org/10.1007/s00453-021-00846-3.
10. Steven Chaplick and Peter Zeman. Combinatorial problems on H-graphs. Electron. Notes Discret. Math., 61:223-229, 2017. URL: https://doi.org/10.1016/j.endm.2017.06.042.
11. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discret. Math., 86(1-3):165-177, 1990. URL: https://doi.org/10.1016/0012-365X(90)90358-O.
12. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
13. Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Martin Milanič. Computing tree decompositions with small independence number. CoRR, abs/2207.09993, 2022. URL: https://arxiv.org/abs/2207.09993.
14. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number in graph classes with a forbidden structure. In Isolde Adler and Haiko Müller, editors, Graph-Theoretic Concepts in Computer Science - 46th International Workshop, WG 2020, Leeds, UK, June 24-26, 2020, Revised Selected Papers, volume 12301 of Lecture Notes in Computer Science, pages 92-105. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-60440-0_8.
15. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. II. tree-independence number. J. Comb. Theory, Ser. B, 164:404-442, 2024. URL: https://doi.org/10.1016/J.JCTB.2023.10.006.
16. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure. Journal of Combinatorial Theory, Series B, 167:338-391, 2024. URL: https://doi.org/10.1016/j.jctb.2024.03.005.
17. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. I. Graph classes with a forbidden structure. SIAM J. Discrete Math., 35(4):2618-2646, 2021. URL: https://doi.org/10.1137/20M1352119.
18. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. A framework for exponential-time-hypothesis-tight algorithms and lower bounds in geometric intersection graphs. SIAM J. Comput., 49(6):1291-1331, 2020. URL: https://doi.org/10.1137/20M1320870.
19. Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. Journal of the ACM, 52(6):866-893, 2005.
20. Irit Dinur. Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover. Electron. Colloquium Comput. Complex., TR16-128, 2016. URL: https://eccc.weizmann.ac.il/report/2016/128, URL: https://arxiv.org/abs/TR16-128.
21. Fedor V. Fomin and Petr A. Golovach. Subexponential parameterized algorithms and kernelization on almost chordal graphs. Algorithmica, 83(7):2170-2214, 2021. URL: https://doi.org/10.1007/s00453-021-00822-x.
22. Fedor V. Fomin, Petr A. Golovach, and Jean-Florent Raymond. On the tractability of optimization problems on H-graphs. Algorithmica, 82(9):2432-2473, 2020. URL: https://doi.org/10.1007/s00453-020-00692-9.
23. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Excluded grid minors and efficient polynomial-time approximation schemes. J. ACM, 65(2):10:1-10:44, 2018. URL: https://doi.org/10.1145/3154833.
24. Esther Galby, Andrea Munaro, and Shizhou Yang. Polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graphs. In Erin W. Chambers and Joachim Gudmundsson, editors, 39th International Symposium on Computational Geometry, SoCG 2023, June 12-15, 2023, Dallas, Texas, USA, volume 258 of LIPIcs, pages 34:1-34:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.34.
25. Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.
26. Martin Grohe. Local tree-width, excluded minors, and approximation algorithms. Combinatorica, 23(4):613-632, 2003. URL: https://doi.org/10.1007/s00493-003-0037-9.
27. Petr Hlinený and Jan Kratochvíl. Representing graphs by disks and balls (a survey of recognition-complexity results). Discret. Math., 229(1-3):101-124, 2001. URL: https://doi.org/10.1016/S0012-365X(00)00204-1.
28. Ashwin Jacob, Fahad Panolan, Venkatesh Raman, and Vibha Sahlot. Structural parameterizations with modulator oblivion. Algorithmica, 84(8):2335-2357, 2022. URL: https://doi.org/10.1007/s00453-022-00971-7.
29. Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, and Kenny Štorgel. Tree decompositions meet induced matchings: beyond Max Weight Independent Set. arXiv:2402.15834, 2024. URL: https://doi.org/10.48550/arXiv.2402.15834.
30. Bingkai Lin. Constant approximating k-clique is W[1]-hard. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1749-1756. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451016.
31. Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Subexponential parameterized algorithms on disk graphs (extended abstract). In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2005-2031. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.80.
32. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense CSPs. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 78:1-78:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.ICALP.2017.78.
33. Dániel Marx. Tractable hypergraph properties for constraint satisfaction and conjunctive queries. J. ACM, 60(6):Art. 42, 51, 2013. URL: https://doi.org/10.1145/2535926.
34. Michal Pilipczuk. Computing tree decompositions. In Fedor V. Fomin, Stefan Kratsch, and Erik Jan van Leeuwen, editors, Treewidth, Kernels, and Algorithms - Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday, volume 12160 of Lecture Notes in Computer Science, pages 189-213. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_14.
35. Vijay Raghavan and Jeremy P. Spinrad. Robust algorithms for restricted domains. J. Algorithms, 48(1):160-172, 2003. URL: https://doi.org/10.1016/S0196-6774(03)00048-8.
36. 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.
37. Neil Robertson and Paul D. Seymour. Graph minors. XIII. The disjoint paths problem. Journal of Combinatorial Theory. Series B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
38. Konstantin Skodinis. Efficient analysis of graphs with small minimal separators. In Peter Widmayer, Gabriele Neyer, and Stephan J. Eidenbenz, editors, Graph-Theoretic Concepts in Computer Science, 25th International Workshop, WG '99, Ascona, Switzerland, June 17-19, 1999, Proceedings, volume 1665 of Lecture Notes in Computer Science, pages 155-166. Springer, 1999. URL: https://doi.org/10.1007/3-540-46784-X_16.
39. Robert E. Tarjan. Decomposition by clique separators. Discrete Math., 55(2):221-232, 1985. URL: https://doi.org/10.1016/0012-365X(85)90051-2.
40. Nikola Yolov. Minor-matching hypertree width. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 219-233. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.16.