A Tight Monte-Carlo Algorithm for Steiner Tree Parameterized by Clique-Width

Authors Narek Bojikian , Stefan Kratsch



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.29.pdf
  • Filesize: 0.88 MB
  • 18 pages

Document Identifiers

Author Details

Narek Bojikian
  • Humboldt-Universität zu Berlin, Germany
Stefan Kratsch
  • Humboldt-Universität zu Berlin, Germany

Cite AsGet BibTex

Narek Bojikian and Stefan Kratsch. A Tight Monte-Carlo Algorithm for Steiner Tree Parameterized by Clique-Width. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.29

Abstract

Given a graph G = (V,E), a set T ⊆ V, and an integer b, the Steiner Tree problem asks whether G has a connected subgraph H with at most b vertices that spans all of T. This work presents a 3^k⋅ n^𝒪(1) time one-sided Monte-Carlo algorithm for solving Steiner Tree when additionally a clique-expression of width k is provided. Known lower bounds for less expressive parameters imply that this dependence on the clique-width of G is optimal assuming the Strong Exponential-Time Hypothesis (SETH). Indeed our work establishes that the parameter dependence of Steiner Tree is the same for any graph parameter between cutwidth and clique-width, assuming SETH. Our work contributes to the program of determining the exact parameterized complexity of fundamental hard problems relative to structural graph parameters such as treewidth, which was initiated by Lokshtanov et al. [SODA 2011 & TALG 2018] and which by now has seen a plethora of results. Since the cut-and-count framework of Cygan et al. [FOCS 2011 & TALG 2022], connectivity problems have played a key role in this program as they pose many challenges for developing tight upper and lower bounds. Recently, Hegerfeld and Kratsch [ESA 2023] gave the first application of the cut-and-count technique to problems parameterized by clique-width and obtained tight bounds for Connected Dominating Set and Connected Vertex Cover, leaving open the complexity of other benchmark connectivity problems such as Steiner Tree and Feedback Vertex Set. Our algorithm for Steiner Tree does not follow the cut-and-count technique and instead works with the connectivity patterns of partial solutions. As a first technical contribution we identify a special family of so-called complete patterns that has strong (existential) representation properties, and using these at least one solution will be preserved. Furthermore, there is a family of 3^k basis patterns that (parity) represents the complete patterns, i.e., it has the same number of solutions modulo two. Our main technical contribution, a new technique called "isolating a representative," allows us to leverage both forms of representation (existential and parity). Both complete patterns and isolation of a representative will likely be applicable to other (connectivity) problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized complexity
  • Steiner tree
  • clique-width

Metrics

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

References

  1. Benjamin Bergougnoux and Mamadou Moustapha Kanté. Fast exact algorithms for some connectivity problems parameterized by clique-width. Theor. Comput. Sci., 782:30-53, 2019. URL: https://doi.org/10.1016/j.tcs.2019.02.030.
  2. Benjamin Bergougnoux and Mamadou Moustapha Kanté. More applications of the d-neighbor equivalence: Acyclicity and connectivity constraints. SIAM J. Discret. Math., 35(3):1881-1926, 2021. URL: https://doi.org/10.1137/20M1350571.
  3. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.008.
  4. Narek Bojikian, Vera Chekan, Falko Hegerfeld, and Stefan Kratsch. Tight bounds for connectivity problems parameterized by cutwidth. In Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, volume 254 of LIPIcs, pages 14:1-14:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.14.
  5. Narek Bojikian and Stefan Kratsch. A tight monte-carlo algorithm for steiner tree parameterized by clique-width. CoRR, abs/2307.14264, 2023. URL: https://doi.org/10.48550/arXiv.2307.14264.
  6. Narek Bojikian and Stefan Kratsch. Tight algorithm for connected odd cycle transversal parameterized by clique-width, 2024. URL: https://arxiv.org/abs/2402.08046.
  7. Glencora Borradaile and Hung Le. Optimal dynamic program for r-domination problems over tree decompositions. In Jiong Guo and Danny Hermelin, editors, 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Denmark, volume 63 of LIPIcs, pages 8:1-8:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.8.
  8. Derek G. Corneil, Michel Habib, Jean-Marc Lanlignel, Bruce A. Reed, and Udi Rotics. Polynomial-time recognition of clique-width ≤3 graphs. Discret. Appl. Math., 160(6):834-865, 2012. URL: https://doi.org/10.1016/j.dam.2011.03.020.
  9. Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci., 46(2):218-270, 1993. URL: https://doi.org/10.1016/0022-0000(93)90004-G.
  10. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  11. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discret. Appl. Math., 101(1-3):77-114, 2000. URL: https://doi.org/10.1016/S0166-218X(99)00184-5.
  12. Radu Curticapean, Nathan Lindzey, and Jesper Nederlof. A tight lower bound for counting hamiltonian cycles via matrix rank. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1080-1099. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.70.
  13. Radu Curticapean and Dániel Marx. Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1650-1669. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch113.
  14. Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast hamiltonicity checking via bases of perfect matchings. J. ACM, 65(3):12:1-12:46, 2018. URL: https://doi.org/10.1145/3148227.
  15. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.23.
  16. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. CoRR, abs/1103.0534, 2011. URL: https://arxiv.org/abs/1103.0534.
  17. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. ACM Trans. Algorithms, 18(2):17:1-17:31, 2022. URL: https://doi.org/10.1145/3506707.
  18. Baris Can Esmer, Jacob Focke, Dániel Marx, and Pawel Rzazewski. List homomorphisms by deleting edges and vertices: tight complexity bounds for bounded-treewidth graphs. CoRR, abs/2210.10677, 2022. URL: https://doi.org/10.48550/arXiv.2210.10677.
  19. Wolfgang Espelage, Frank Gurski, and Egon Wanke. How to solve np-hard graph problems on clique-width bounded graphs in polynomial time. In Andreas Brandstädt and Van Bang Le, editors, Graph-Theoretic Concepts in Computer Science, 27th International Workshop, WG 2001, Boltenhagen, Germany, June 14-16, 2001, Proceedings, volume 2204 of Lecture Notes in Computer Science, pages 117-128. Springer, 2001. URL: https://doi.org/10.1007/3-540-45477-2_12.
  20. Michael R. Fellows, Frances A. Rosamond, Udi Rotics, and Stefan Szeider. Clique-width is np-complete. SIAM J. Discret. Math., 23(2):909-939, 2009. URL: https://doi.org/10.1137/070687256.
  21. Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp Schepper, and Philip Wellnitz. Tight complexity bounds for counting generalized dominating sets in bounded-treewidth graphs. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3664-3683. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch140.
  22. Jacob Focke, Dániel Marx, and Pawel Rzazewski. Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 431-458. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.22.
  23. Robert Ganian, Thekla Hamm, Viktoriia Korchemna, Karolina Okrasa, and Kirill Simonov. The fine-grained complexity of graph homomorphism parameterized by clique-width. In Mikolaj Bojanczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming, ICALP 2022, July 4-8, 2022, Paris, France, volume 229 of LIPIcs, pages 66:1-66:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.66.
  24. Carla Groenland, Isja Mannens, Jesper Nederlof, and Krisztina Szilágyi. Tight bounds for counting colorings and connected edge sets parameterized by cutwidth. In Petra Berenbrink and Benjamin Monmege, editors, 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15-18, 2022, Marseille, France (Virtual Conference), volume 219 of LIPIcs, pages 36:1-36:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.36.
  25. Frank Gurski and Egon Wanke. Vertex disjoint paths on clique-width bounded graphs. Theor. Comput. Sci., 359(1-3):188-199, 2006. URL: https://doi.org/10.1016/j.tcs.2006.02.026.
  26. Tesshu Hanaka, Yasuaki Kobayashi, and Taiga Sone. A (probably) optimal algorithm for bisection on bounded-treewidth graphs. Theor. Comput. Sci., 873:38-46, 2021. URL: https://doi.org/10.1016/j.tcs.2021.04.023.
  27. Falko Hegerfeld and Stefan Kratsch. Tight algorithms for connectivity problems parameterized by clique-width. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 59:1-59:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.59.
  28. Falko Hegerfeld and Stefan Kratsch. Tight algorithms for connectivity problems parameterized by modular-treewidth. In Daniël Paulusma and Bernard Ries, editors, Graph-Theoretic Concepts in Computer Science - 49th International Workshop, WG 2023, Fribourg, Switzerland, June 28-30, 2023, Revised Selected Papers, volume 14093 of Lecture Notes in Computer Science, pages 388-402. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-43380-1_28.
  29. Yoichi Iwata and Yuichi Yoshida. On the equivalence among problems of bounded width. 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 754-765. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_63.
  30. Bart M. P. Jansen and Jesper Nederlof. Computing the chromatic number using graph decompositions via matrix rank. Theor. Comput. Sci., 795:520-539, 2019. URL: https://doi.org/10.1016/j.tcs.2019.08.006.
  31. Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structural parameters, tight bounds, and approximation for (k, r)-center. Discret. Appl. Math., 264:90-117, 2019. URL: https://doi.org/10.1016/j.dam.2018.11.002.
  32. Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structurally parameterized d-scattered set. Discret. Appl. Math., 308:168-186, 2022. URL: https://doi.org/10.1016/j.dam.2020.03.052.
  33. Michael Lampis. Model checking lower bounds for simple graphs. In Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, volume 7965 of Lecture Notes in Computer Science, pages 673-683. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-39206-1_57.
  34. Michael Lampis. Finer tight bounds for coloring on clique-width. SIAM J. Discret. Math., 34(3):1538-1558, 2020. URL: https://doi.org/10.1137/19M1280326.
  35. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs on bounded treewidth are probably optimal. In Dana Randall, editor, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 777-789. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.61.
  36. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018. URL: https://doi.org/10.1145/3170442.
  37. Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and gaps: Tight complexity results of general factor problems parameterized by treewidth and cutwidth. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 95:1-95:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.95.
  38. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Comb., 7(1):105-113, 1987. URL: https://doi.org/10.1007/BF02579206.
  39. Jesper Nederlof. Algorithms for np-hard problems via rank-related parameters of matrices. 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 145-164. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_11.
  40. Karolina Okrasa, Marta Piecyk, and Pawel Rzazewski. Full complexity classification of the list homomorphism problem for bounded-treewidth graphs. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 74:1-74:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.74.
  41. Karolina Okrasa and Pawel Rzazewski. Fine-grained complexity of the graph homomorphism problem for bounded-treewidth graphs. SIAM J. Comput., 50(2):487-508, 2021. URL: https://doi.org/10.1137/20M1320146.
  42. Sang-il Oum. Approximating rank-width and clique-width quickly. ACM Trans. Algorithms, 5(1):10:1-10:20, 2008. URL: https://doi.org/10.1145/1435375.1435385.
  43. Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width. J. Comb. Theory, Ser. B, 96(4):514-528, 2006. URL: https://doi.org/10.1016/j.jctb.2005.10.006.
  44. Marta Piecyk and Pawel Rzazewski. Fine-grained complexity of the list homomorphism problem: Feedback vertex set and cutwidth. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021, March 16-19, 2021, Saarbrücken, Germany (Virtual Conference), volume 187 of LIPIcs, pages 56:1-56:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.56.
  45. Bas A. M. van Geffen, Bart M. P. Jansen, Arnoud A. W. M. de Kroon, and Rolf Morel. Lower bounds for dynamic programming on planar graphs of bounded cutwidth. J. Graph Algorithms Appl., 24(3):461-482, 2020. URL: https://doi.org/10.7155/jgaa.00542.
  46. Egon Wanke. k-nlc graphs and polynomial algorithms. Discret. Appl. Math., 54(2-3):251-266, 1994. URL: https://doi.org/10.1016/0166-218X(94)90026-4.