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Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra

Authors Václav Blažej , Dušan Knop , Jan Pokorný , Šimon Schierreich

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Author Details

Václav Blažej
  • University of Warwick, UK
Dušan Knop
  • Czech Technical University in Prague, Czech Republic
Jan Pokorný
  • Czech Technical University in Prague, Czech Republic
Šimon Schierreich
  • Czech Technical University in Prague, Czech Republic

Funding

This work was co-funded by the European Union under the project Robotics and advanced industrial production (reg. no. CZ.02.01.01/00/22_008/0004590).
  • Blažej, Václav: Supported by the Engineering and Physical Sciences Research Council [grant number EP/V044621/1].
  • Knop, Dušan: Acknowledges the support of the Czech Science Foundation Grant No. 22-19557S.
  • Pokorný, Jan: Acknowledges the support of the Grant Agency of the CTU in Prague funded grant No. SGS23/205/OHK3/3T/18.
  • Schierreich, Šimon: Acknowledges the support of the Czech Science Foundation Grant No. 22-19557S and of the Grant Agency of the CTU in Prague funded grant No. SGS23/205/OHK3/3T/18.

Acknowledgements

We are grateful to all anonymous referees for their valuable comments.

Cite AsGet BibTex

Václav Blažej, Dušan Knop, Jan Pokorný, and Šimon Schierreich. Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.29

Abstract

In the Equitable Connected Partition (ECP for short) problem, we are given a graph G = (V,E) together with an integer p ∈ ℕ, and our goal is to find a partition of V into p parts such that each part induces a connected sub-graph of G and the size of each two parts differs by at most 1. On the one hand, the problem is known to be NP-hard in general and W[1]-hard with respect to the path-width, the feedback-vertex set, and the number of parts p combined. On the other hand, fixed-parameter algorithms are known for parameters the vertex-integrity and the max leaf number. In this work, we systematically study ECP with respect to various structural restrictions of the underlying graph and provide a clear dichotomy of its parameterised complexity. Specifically, we show that the problem is in FPT when parameterized by the modular-width and the distance to clique. Next, we prove W[1]-hardness with respect to the distance to cluster, the 4-path vertex cover number, the distance to disjoint paths, and the feedback-edge set, and NP-hardness for constant shrub-depth graphs. Our hardness results are complemented by matching algorithmic upper-bounds: we give an XP algorithm for parameterisation by the tree-width and the distance to cluster. We also give an improved FPT algorithm for parameterisation by the vertex integrity and the first explicit FPT algorithm for the 3-path vertex cover number. The main ingredient of these algorithms is a formulation of ECP as N-fold IP, which clearly indicates that such formulations may, in certain scenarios, significantly outperform existing algorithms based on the famous algorithm of Lenstra.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Equitable Connected Partition
  • structural parameters
  • fixed-parameter tractability
  • N-fold integer programming
  • tree-width
  • shrub-depth
  • modular-width

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References

  1. Akanksha Agrawal and M. S. Ramanujan. Distance from triviality 2.0: Hybrid parameterizations. In Cristina Bazgan and Henning Fernau, editors, Proceedings of the 33rd International Workshop on Combinatorial Algorithms, IWOCA '22, volume 13270 of Lecture Notes in Computer Science, pages 3-20. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-06678-8_1.
  2. Micah Altman. The computational complexity of automated redistricting: Is automation the answer? Rutgers Computer and Law Technology Journal, 23(1):81-142, March 1997. Google Scholar
  3. Kateřina Altmanová, Dušan Knop, and Martin Koutecký. Evaluating and tuning n-fold integer programming. ACM Journal of Experimental Algorithmics, 24(2), July 2019. URL: https://doi.org/10.1145/3330137.
  4. Konstantin Andreev and Harald Racke. Balanced graph partitioning. Theory of Computing Systems, 39(6):929-939, November 2006. URL: https://doi.org/10.1007/s00224-006-1350-7.
  5. Peter Arbenz, G. Harry van Lenthe, Uche Mennel, Ralph Müller, and Marzio Sala. Multi-level μ-finite element analysis for human bone structures. In Bo Kågström, Erik Elmroth, Jack Dongarra, and Jerzy Waśniewski, editors, Proceedings of the 8th International Workshop on Applied Parallel Computing, PARA '06, volume 4699 of Lecture Notes in Computer Science, pages 240-250. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-75755-9_30.
  6. René van Bevern, Andreas Emil Feldmann, Manuel Sorge, and Ondřej Suchý. On the parameterized complexity of computing balanced partitions in graphs. Theory of Computing Systems, 57:1-35, July 2015. URL: https://doi.org/10.1007/s00224-014-9557-5.
  7. Sandeep N. Bhatt and Frank Thomson Leighton. A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences, 28(2):300-343, April 1984. URL: https://doi.org/10.1016/0022-0000(84)90071-0.
  8. Václav Blažej, Robert Ganian, Dušan Knop, Jan Pokorný, Šimon Schierreich, and Kirill Simonov. The parameterized complexity of network microaggregation. In Proceedings of the 37th AAAI Conference on Artificial Intelligence, AAAI '23, pages 6262-6270. AAAI Press, 2023. URL: https://doi.org/10.1609/aaai.v37i5.25771.
  9. Hans L. Bodlaender and Fedor V. Fomin. Equitable colorings of bounded treewidth graphs. Theoretical Computer Science, 349(1):22-30, December 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.027.
  10. Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, Stéphan Thomassé, and Rémi Watrigant. Twin-width and polynomial kernels. Algorithmica, 84(11):3300-3337, 2022. URL: https://doi.org/10.1007/S00453-022-00965-5.
  11. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: Tractable FO model checking. Journal of the ACM, 69(1), 2022. URL: https://doi.org/10.1145/3486655.
  12. Robert Bredereck, Andrzej Kaczmarczyk, Dušan Knop, and Rolf Niedermeier. High-multiplicity fair allocation: Lenstra empowered by N-fold integer programming. In Anna Karlin, Nicole Immorlica, and Ramesh Johari, editors, Proceedings of the 20th ACM Conference on Economics and Computation, EC '19, pages 505-523. ACM, 2019. URL: https://doi.org/10.1145/3328526.3329649.
  13. Thang Nguyen Bui and Andrew Peck. Partitioning planar graphs. SIAM Journal on Computing, 21(2):203-215, 1992. URL: https://doi.org/10.1137/0221016.
  14. Aydın Buluç, Henning Meyerhenke, Ilya Safro, Peter Sanders, and Christian Schulz. Recent advances in graph partitioning. In Lasse Kliemann and Peter Sanders, editors, Algorithm Engineering: Selected Results and Surveys, pages 117-158. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-49487-6_4.
  15. Bor-Liang Chen, Ming-Tat Ko, and Ko-Wei Lih. Equitable and m-bounded coloring of split graphs. In Michel Deza, Reinhardt Euler, and Ioannis Manoussakis, editors, Combinatorics and Computer Science, CCS '95, volume 1120 of Lecture Notes in Computer Science, pages 1-5. Springer, 1996. Google Scholar
  16. Bor-Liang Chen and Ko-Wei Lih. Equitable coloring of trees. Journal of Combinatorial Theory, Series B, 61(1):83-87, May 1994. Google Scholar
  17. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1):77-114, 2000. URL: https://doi.org/10.1016/S0166-218X(99)00184-5.
  18. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, Cham, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  19. Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Minimum bisection is fixed-parameter tractable. SIAM Journal on Computing, 48(2):417-450, 2019. URL: https://doi.org/10.1137/140988553.
  20. Argyrios Deligkas, Eduard Eiben, Robert Ganian, Thekla Hamm, and Sebastian Ordyniak. The parameterized complexity of connected fair division. In Zhi-Hua Zhou, editor, Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI '21, pages 139-145. International Joint Conferences on Artificial Intelligence Organization, August 2021. Main Track. URL: https://doi.org/10.24963/ijcai.2021/20.
  21. Reinhard Diestel. Graph Theory. Graduate Texts in Mathematics. Springer, Berlin, Heidelberg, 5th edition, 2017. URL: https://doi.org/10.1007/978-3-662-53622-3.
  22. Martin Doucha and Jan Kratochvíl. Cluster vertex deletion: A parameterization between vertex cover and clique-width. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science, MFCS '12, volume 7464 of Lecture Notes in Computer Science, pages 348-359. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_32.
  23. M. E. Dyer and A. M. Frieze. A partitioning algorithm for minimum weighted Euclidean matching. Information Processing Letters, 18(2):59-62, 1984. URL: https://doi.org/10.1016/0020-0190(84)90024-3.
  24. Josep Díaz and George B. Mertzios. Minimum bisection is NP-hard on unit disk graphs. Information and Computation, 256:83-92, 2017. URL: https://doi.org/10.1016/j.ic.2017.04.010.
  25. Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn. An algorithmic theory of integer programming. CoRR, abs/1904.01361, 2019. URL: https://arxiv.org/abs/1904.01361.
  26. Rosa Enciso, Michael R. Fellows, Jiong Guo, Iyad Kanj, Frances Rosamond, and Ondřej Suchý. What makes equitable connected partition easy. In Jianer Chen and Fedor V. Fomin, editors, Proceedings of the 4th International Workshop on Parameterized and Exact Computation, IWPEC '09, volume 5917 of Lecture Notes in Computer Science, pages 122-133. Springer, 2009. Google Scholar
  27. Andreas Emil Feldmann and Luca Foschini. Balanced partitions of trees and applications. Algorithmica, 71(2):354-376, February 2015. URL: https://doi.org/10.1007/s00453-013-9802-3.
  28. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances Rosamond, Saket Saurabh, Stefan Szeider, and Carsten Thomassen. On the complexity of some colorful problems parameterized by treewidth. Information and Computation, 209(2):143-153, 2011. URL: https://doi.org/10.1016/j.ic.2010.11.026.
  29. Jiří Fiala, Petr A. Golovach, and Jan Kratochvíl. Parameterized complexity of coloring problems: Treewidth versus vertex cover. Theoretical Computer Science, 412(23):2513-2523, 2011. URL: https://doi.org/10.1016/j.tcs.2010.10.043.
  30. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM Journal on Computing, 43(5):1541-1563, 2014. URL: https://doi.org/10.1137/130910932.
  31. H. Furmańczyk and M. Kubale. The complexity of equitable vertex coloring of graphs. Journal of Applied Computer Science, 13(2):95-106, 2005. Google Scholar
  32. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In Gregory Gutin and Stefan Szeider, editors, Proceedings of the 8th International Symposium on Parameterized and Exact Computation, IPEC '13, volume 8246 of Lecture Notes in Computer Science, pages 163-176. Springer, 2013. Google Scholar
  33. Robert Ganian, Petr Hliněný, Jaroslav Nešetřil, Jan Obdržálek, and Patrice Ossona de Mendez. Shrub-depth: Capturing height of dense graphs. Logical Methods in Computer Science, 15(1), 2019. URL: https://doi.org/10.23638/LMCS-15(1:7)2019.
  34. Robert Ganian, Petr Hliněný, Jaroslav Nešetřil, Jan Obdržálek, Patrice Ossona de Mendez, and Reshma Ramadurai. When trees grow low: Shrubs and fast MSO1. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science, MFCS '12, volume 7464 of Lecture Notes in Computer Science, pages 419-430. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_38.
  35. Robert Ganian and Jan Obdržálek. Expanding the expressive power of monadic second-order logic on restricted graph classes. In Thierry Lecroq and Laurent Mouchard, editors, Proceedings of the 24th International Workshop on Combinatorial Algorithms, IWOCA '13, volume 8288 of Lecture Notes in Computer Science, pages 164-177. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-45278-9_15.
  36. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., 1979. Google Scholar
  37. Tomáš Gavenčiak, Martin Koutecký, and Dušan Knop. Integer programming in parameterized complexity: Five miniatures. Discrete Optimization, 44:100596, 2022. Optimization and Discrete Geometry. URL: https://doi.org/10.1016/j.disopt.2020.100596.
  38. Tatsuya Gima, Tesshu Hanaka, Masashi Kiyomi, Yasuaki Kobayashi, and Yota Otachi. Exploring the gap between treedepth and vertex cover through vertex integrity. Theoretical Computer Science, 918:60-76, 2022. URL: https://doi.org/10.1016/j.tcs.2022.03.021.
  39. Tatsuya Gima and Yota Otachi. Extended MSO model checking via small vertex integrity. Algorithmica, 86(1):147-170, 2024. URL: https://doi.org/10.1007/S00453-023-01161-9.
  40. Guilherme C. M. Gomes, Matheus R. Guedes, and Vinicius F. dos Santos. Structural parameterizations for equitable coloring: Complexity, FPT algorithms, and kernelization. Algorithmica, 85:1912-1947, July 2023. URL: https://doi.org/10.1007/s00453-022-01085-w.
  41. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Rod Downey, Michael Fellows, and Frank Dehne, editors, Proceedings of the 1st International Workshop on Parameterized and Exact Computation, IWPEC '04, pages 162-173. Springer, 2004. Google Scholar
  42. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  43. Takehiro Ito, Xiao Zhou, and Takao Nishizeki. Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size. Journal of Discrete Algorithms, 4(1):142-154, 2006. URL: https://doi.org/10.1016/j.jda.2005.01.005.
  44. Klaus Jansen, Stefan Kratsch, Dániel Marx, and Ildikó Schlotter. Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences, 79(1):39-49, 2013. URL: https://doi.org/10.1016/j.jcss.2012.04.004.
  45. Dušan Knop, Martin Koutecký, Asaf Levin, Matthias Mnich, and Shmuel Onn. High-multiplicity N-fold IP via configuration LP. Mathematical Programming, 200(1):199-227, 2023. URL: https://doi.org/10.1007/s10107-022-01882-9.
  46. Dušan Knop, Martin Koutecký, and Matthias Mnich. Combinatorial N-fold integer programming and applications. Mathematical Programming, 184(1):1-34, 2020. URL: https://doi.org/10.1007/s10107-019-01402-2.
  47. Dušan Knop, Šimon Schierreich, and Ondřej Suchý. Balancing the spread of two opinions in sparse social networks (student abstract). In Proceedings of the 36th AAAI Conference on Artificial Intelligence, AAAI '22, pages 12987-12988. AAAI Press, 2022. URL: https://doi.org/10.1609/aaai.v36i11.21630.
  48. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, September 2012. URL: https://doi.org/10.1007/s00453-011-9554-x.
  49. Hendrik W. Lenstra, Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4), 1983. URL: https://doi.org/10.1287/moor.8.4.538.
  50. Harry A. Levin and Sorelle A. Friedler. Automated congressional redistricting. ACM Journal of Experimental Algorithmics, 24, April 2019. URL: https://doi.org/10.1145/3316513.
  51. Ko-Wei Lih. Equitable coloring of graphs. In Panos M. Pardalos, Ding-Zhu Du, and Ronald L. Graham, editors, Handbook of Combinatorial Optimization, pages 1199-1248. Springer, 2013. URL: https://doi.org/10.1007/978-1-4419-7997-1_25.
  52. Mario Lucertini, Yehoshua Perl, and Bruno Simeone. Most uniform path partitioning and its use in image processing. Discrete Applied Mathematics, 42(2):227-256, 1993. URL: https://doi.org/10.1016/0166-218X(93)90048-S.
  53. Kitty Meeks and Fiona Skerman. The parameterised complexity of computing the maximum modularity of a graph. Algorithmica, 82(8):2174-2199, August 2020. URL: https://doi.org/10.1007/s00453-019-00649-7.
  54. Manfred Wiegers. The k-section of treewidth restricted graphs. In Branislav Rovan, editor, Proceedings of the 15th International Symposium on Mathematical Foundations of Computer Science, MFCS '90, volume 452 of Lecture Notes in Computer Science, pages 530-537. Springer, 1990. Google Scholar
  55. Justin C. Williams Jr. Political redistricting: A review. Papers in Regional Science, 74(1), 1995. URL: https://doi.org/10.1111/j.1435-5597.1995.tb00626.x.
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