Document Open Access Logo

True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs

Authors Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, Jie Xue

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2022.11.pdf
  • Filesize: 0.88 MB
  • 16 pages

Document Identifiers

Author Details

Sayan Bandyapadhyay
  • University of Bergen, Norway
William Lochet
  • LIRMM, Université de Montpellier, CNRS, France
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
Jie Xue
  • New York University Shanghai, China

Funding

  • Lokshtanov, Daniel: Supported by BSF award 2018302 and NSF award CCF-2008838.
  • Saurabh, Saket: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416), and Swarnajayanti Fellowship (No. DST/SJF/MSA01/2017-18).

Cite AsGet BibTex

Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, and Jie Xue. True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.11

Abstract

We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set 𝒟 of n unit disks inducing a unit-disk graph G_𝒟 and a number p ∈ [n], one can partition 𝒟 into p subsets 𝒟₁,… ,𝒟_p such that for every i ∈ [p] and every 𝒟' ⊆ 𝒟_i, the graph obtained from G_𝒟 by contracting all edges between the vertices in 𝒟_i $1𝒟' admits a tree decomposition in which each bag consists of O(p+|𝒟'|) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved very recently by Marx et al. [SODA'22] and Bandyapadhyay et al. [SODA'22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem (CDT) for unit-disk graphs, resolving an open question in the work Panolan et al. [SODA'19]. On the algorithmic side, we obtain a new FPT algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in 2^{O(√k log k)} ⋅ n^{O(1)} time, where k denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA'22] (which works more generally for disk graphs) and is almost optimal, as the problem cannot be solved in 2^{o(√k)} ⋅ n^{O(1)} time assuming the ETH.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • unit-disk graphs
  • tree decomposition
  • contraction decomposition
  • bipartization

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. o (√log n) approximation algorithms for min uncut, min 2cnf deletion, and directed cut problems. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 573-581, 2005. Google Scholar
  2. Jochen Alber, Henning Fernau, and Rolf Niedermeier. Graph separators: a parameterized view. J. Comput. Syst. Sci., 67(4):808-832, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00072-2.
  3. Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004. URL: https://doi.org/10.1016/j.jalgor.2003.10.001.
  4. Shinwoo An and Eunjin Oh. Feedback vertex set on geometric intersection graphs. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 47:1-47:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.47.
  5. Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, and Jie Xue. Subexponential parameterized algorithms for cut and cycle hitting problems on h-minor-free graphs. CoRR, to appear in SODA 2022, abs/2111.14196, 2021. URL: http://arxiv.org/abs/2111.14196.
  6. Thang Nguyen Bui and Andrew Peck. Partitioning planar graphs. SIAM J. Comput., 21(2):203-215, 1992. URL: https://doi.org/10.1137/0221016.
  7. Sergio Cabello and Miha Jejčič. Shortest paths in intersection graphs of unit disks. Computational Geometry, 48(4):360-367, 2015. Google Scholar
  8. Timothy M Chan and Dimitrios Skrepetos. All-pairs shortest paths in geometric intersection graphs. In Workshop on Algorithms and Data Structures, pages 253-264. Springer, 2017. Google Scholar
  9. Timothy M Chan and Dimitrios Skrepetos. Approximate shortest paths and distance oracles in weighted unit-disk graphs. Journal of Computational Geometry (Old Web Site), 10(2):3-20, 2019. Google Scholar
  10. Brent N Clark, Charles J Colbourn, and David S Johnson. Unit disk graphs. Discrete mathematics, 86(1-3):165-177, 1990. Google Scholar
  11. 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.
  12. Mark de Berg, Hans L Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C van der Zanden. A framework for eth-tight algorithms and lower bounds in geometric intersection graphs. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 574-586, 2018. Google Scholar
  13. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, 2005. URL: https://doi.org/10.1145/1101821.1101823.
  14. Erik D. Demaine, MohammadTaghi Hajiaghayi, and Ken-ichi Kawarabayashi. Contraction decomposition in h-minor-free graphs and algorithmic applications. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 441-450, 2011. Google Scholar
  15. Erik D. Demaine, MohammadTaghi Hajiaghayi, and Bojan Mohar. Approximation algorithms via contraction decomposition. Combinatorica, 30(5):533-552, 2010. Google Scholar
  16. Frederic Dorn, Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs. Inf. Comput., 233:60-70, 2013. URL: https://doi.org/10.1016/j.ic.2013.11.006.
  17. Samuel Fiorini, Nadia Hardy, Bruce Reed, and Adrian Vetta. Planar graph bipartization in linear time. Discrete Applied Mathematics, 156(7):1175-1180, 2008. Google Scholar
  18. Fedor V. Fomin, Daniel Lokshtanov, Sudeshna Kolay, Fahad Panolan, and Saket Saurabh. Subexponential algorithms for rectilinear steiner tree and arborescence problems. ACM Trans. Algorithms, 16(2):21:1-21:37, 2020. URL: https://doi.org/10.1145/3381420.
  19. Fedor V. Fomin, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Subexponential parameterized algorithms for planar and apex-minor-free graphs via low treewidth pattern covering. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 515-524. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.62.
  20. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of map graphs with applications. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 60:1-60:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.60.
  21. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, hitting and packing cycles in subexponential time on unit disk graphs. Discret. Comput. Geom., 62(4):879-911, 2019. URL: https://doi.org/10.1007/s00454-018-00054-x.
  22. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Eth-tight algorithms for long path and cycle on unit disk graphs. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry, SoCG 2020, June 23-26, 2020, Zürich, Switzerland, volume 164 of LIPIcs, pages 44:1-44:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.44.
  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. Jie Gao and Li Zhang. Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM Journal on Computing, 35(1):151-169, 2005. Google Scholar
  25. Michel X Goemans and David P Williamson. Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica, 18(1):37-59, 1998. Google Scholar
  26. MohammadTaghi Hajiaghayi. Contraction and minor graph decomposition and their algorithmic applications. Filmed Talk at Microsoft Research, 2016. Google Scholar
  27. William K Hale. Frequency assignment: Theory and applications. Proceedings of the IEEE, 68(12):1497-1514, 1980. Google Scholar
  28. Falk Hüffner. Algorithm engineering for optimal graph bipartization. Journal of Graph Algorithms and Applications, 13(2):77-98, 2009. Google Scholar
  29. Bart M. P. Jansen, Marcin Pilipczuk, and Erik Jan van Leeuwen. A deterministic polynomial kernel for odd cycle transversal and vertex multiway cut in planar graphs. In Rolf Niedermeier and Christophe Paul, editors, 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany, volume 126 of LIPIcs, pages 39:1-39:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.39.
  30. Ken-ichi Kawarabayashi and Bruce Reed. An (almost) linear time algorithm for odd cycles transversal. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 365-378. SIAM, 2010. Google Scholar
  31. Philip N. Klein. A subset spanner for planar graphs, : with application to subset TSP. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 749-756, 2006. URL: https://doi.org/10.1145/1132516.1132620.
  32. Philip N. Klein. A linear-time approximation scheme for TSP in undirected planar graphs with edge-weights. SIAM J. Comput., 37(6):1926-1952, 2008. URL: https://doi.org/10.1137/060649562.
  33. Philip N. Klein and Dániel Marx. Solving planar k -terminal cut in O(n^c√k) time. In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, volume 7391 of Lecture Notes in Computer Science, pages 569-580. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31594-7_48.
  34. Philip N. Klein and Dániel Marx. A subexponential parameterized algorithm for subset TSP on planar graphs. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1812-1830. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.131.
  35. Stefan Kratsch and Magnus Wahlström. Compression via matroids: A randomized polynomial kernel for odd cycle transversal. ACM Trans. Algorithms, 10(4):20:1-20:15, 2014. URL: https://doi.org/10.1145/2635810.
  36. Daniel Lokshtanov, NS Narayanaswamy, Venkatesh Raman, MS Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms (TALG), 11(2):1-31, 2014. Google Scholar
  37. Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Subexponential parameterized algorithms on disk graphs. to appear in SODA 2022, 2021. Google Scholar
  38. Daniel Lokshtanov, Saket Saurabh, and Magnus Wahlström. Subexponential parameterized odd cycle transversal on planar graphs. In Deepak D'Souza, Telikepalli Kavitha, and Jaikumar Radhakrishnan, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, volume 18 of LIPIcs, pages 424-434. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2012.424.
  39. Dániel Marx, Pranabendu Misra, Daniel Neuen, and Prafullkumar Tale. A framework for parameterized subexponential algorithms for generalized cycle hitting problems on planar graphs. CoRR, to appear in SODA 2022, abs/2110.15098, 2021. URL: http://arxiv.org/abs/2110.15098.
  40. Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. On subexponential parameterized algorithms for steiner tree and directed subset TSP on planar graphs. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 474-484. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00052.
  41. Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 865-877. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_72.
  42. Jesper Nederlof. Detecting and counting small patterns in planar graphs in subexponential parameterized time. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 1293-1306. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384261.
  43. Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Contraction decomposition in unit disk graphs and algorithmic applications in parameterized complexity. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1035-1054. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.64.
  44. Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Parameterized computational geometry via decomposition theorems. In Gautam K. Das, Partha Sarathi Mandal, Krishnendu Mukhopadhyaya, and Shin-Ichi Nakano, editors, WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Guwahati, India, February 27 - March 2, 2019, Proceedings, volume 11355 of Lecture Notes in Computer Science, pages 15-27. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-10564-8_2.
  45. Bruce Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Operations Research Letters, 32(4):299-301, 2004. Google Scholar
  46. Siamak Tazari. Faster approximation schemes and parameterized algorithms on (odd-)h-minor-free graphs. Theor. Comput. Sci., 417:95-107, 2012. URL: https://doi.org/10.1016/j.tcs.2011.09.014.
  47. Haitao Wang and Jie Xue. Near-optimal algorithms for shortest paths in weighted unit-disk graphs. Discrete & Computational Geometry, 64(4):1141-1166, 2020. Google Scholar
  48. Mihalis Yannakakis. Node-and edge-deletion np-complete problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, pages 253-264, 1978. Google Scholar
  49. Yu-Shuan Yeh, Joanne C Wilson, and Stuart C Schwartz. Outage probability in mobile telephony with directive antennas and macrodiversity. IEEE transactions on vehicular technology, 33(3):123-127, 1984. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact