Papers
Topics
Authors
Recent
Search
2000 character limit reached

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

Published 29 Apr 2024 in cs.DS | (2404.18968v1)

Abstract: We study the Equitable Connected Partition (ECP for short) problem, where we are given a graph G=(V,E) together with an integer p, 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. As our main contribution, we resolve a long-standing open question [Enciso et al.; IWPEC '09] regarding the parameterisation by the tree-depth of the underlying graph. In particular, we show that ECP is W[1]-hard with respect to the 4-path vertex cover number, which is an even more restrictive structural parameter than the tree-depth. In addition to that, we show W[1]-hardness of the problem with respect to the feedback-edge set, the distance to disjoint paths, and NP-hardness with respect to the shrub-depth and the clique-width. On a positive note, we propose several novel fixed-parameter algorithms for various parameters that are bounded for dense graphs.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (60)
  1. 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. doi: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.
  3. Evaluating and tuning n𝑛nitalic_n-fold integer programming. ACM Journal of Experimental Algorithmics, 24(2), July 2019. doi:10.1145/3330137.
  4. Balanced graph partitioning. Theory of Computing Systems, 39(6):929–939, November 2006. doi:10.1007/s00224-006-1350-7.
  5. Multi-level μ𝜇\muitalic_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. doi:10.1007/978-3-540-75755-9_30.
  6. Improved bounds on Bell numbers and on moments of sums of random variables. Probability and Mathematical Statistics, 30(2):185–205, 2000.
  7. On the parameterized complexity of computing balanced partitions in graphs. Theory of Computing Systems, 57:1–35, July 2015. doi:10.1007/s00224-014-9557-5.
  8. 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. doi:10.1016/0022-0000(84)90071-0.
  9. The parameterized complexity of network microaggregation. In Proceedings of the 37th AAAI Conference on Artificial Intelligence, AAAI ’23, volume 37, part 5, pages 6262–6270. AAAI Press, 2023. doi:10.1609/aaai.v37i5.25771.
  10. Equitable colorings of bounded treewidth graphs. Theoretical Computer Science, 349(1):22–30, December 2005. doi:10.1016/j.tcs.2005.09.027.
  11. Twin-width and polynomial kernels. Algorithmica, 84(11):3300–3337, 2022. doi:10.1007/S00453-022-00965-5.
  12. Twin-width I: Tractable FO model checking. Journal of the ACM, 69(1), 2022. doi:10.1145/3486655.
  13. 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. doi:10.1145/3328526.3329649.
  14. Partitioning planar graphs. SIAM Journal on Computing, 21(2):203–215, 1992. doi:10.1137/0221016.
  15. Recent advances in graph partitioning. In Lasse Kliemann and Peter Sanders, editors, Algorithm Engineering: Selected Results and Surveys, pages 117–158. Springer, 2016. doi:10.1007/978-3-319-49487-6_4.
  16. Equitable and m𝑚mitalic_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.
  17. Equitable coloring of trees. Journal of Combinatorial Theory, Series B, 61(1):83–87, May 1994.
  18. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1):77–114, 2000. doi:10.1016/S0166-218X(99)00184-5.
  19. Parameterized Algorithms. Springer, Cham, 2015. doi:10.1007/978-3-319-21275-3.
  20. Minimum bisection is fixed-parameter tractable. SIAM Journal on Computing, 48(2):417–450, 2019. doi:10.1137/140988553.
  21. 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. doi:10.24963/ijcai.2021/20.
  22. Reinhard Diestel. Graph Theory. Graduate Texts in Mathematics. Springer, Berlin, Heidelberg, 5th edition, 2017. doi:10.1007/978-3-662-53622-3.
  23. 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. doi:10.1007/978-3-642-32589-2_32.
  24. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London, 2013. doi:10.1007/978-1-4471-5559-1.
  25. A partitioning algorithm for minimum weighted Euclidean matching. Information Processing Letters, 18(2):59–62, 1984. doi:10.1016/0020-0190(84)90024-3.
  26. Minimum bisection is NP-hard on unit disk graphs. Information and Computation, 256:83–92, 2017. doi:10.1016/j.ic.2017.04.010.
  27. An algorithmic theory of integer programming. CoRR, abs/1904.01361, 2019. URL: http://arxiv.org/abs/1904.01361, arXiv:1904.01361.
  28. 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.
  29. Balanced partitions of trees and applications. Algorithmica, 71(2):354–376, February 2015. doi:10.1007/s00453-013-9802-3.
  30. On the complexity of some colorful problems parameterized by treewidth. Information and Computation, 209(2):143–153, 2011. doi:10.1016/j.ic.2010.11.026.
  31. Parameterized complexity of coloring problems: Treewidth versus vertex cover. Theoretical Computer Science, 412(23):2513–2523, 2011. doi:10.1016/j.tcs.2010.10.043.
  32. Almost optimal lower bounds for problems parameterized by clique-width. SIAM Journal on Computing, 43(5):1541–1563, 2014. doi:10.1137/130910932.
  33. H. Furmańczyk and M. Kubale. The complexity of equitable vertex coloring of graphs. Journal of Applied Computer Science, 13(2):95–106, 2005.
  34. 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.
  35. Robert Ganian. Improving vertex cover as a graph parameter. Discrete Mathematics & Theoretical Computer Science, 17(2):77–100, 2015. doi:10.46298/DMTCS.2136.
  36. Shrub-depth: Capturing height of dense graphs. Logical Methods in Computer Science, 15(1), 2019. doi:10.23638/LMCS-15(1:7)2019.
  37. 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. doi:10.1007/978-3-642-32589-2_38.
  38. 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. doi:10.1007/978-3-642-45278-9_15.
  39. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., 1979.
  40. Integer programming in parameterized complexity: Five miniatures. Discrete Optimization, 44:100596, 2022. Optimization and Discrete Geometry. doi:10.1016/j.disopt.2020.100596.
  41. Exploring the gap between treedepth and vertex cover through vertex integrity. Theoretical Computer Science, 918:60–76, 2022. doi:10.1016/j.tcs.2022.03.021.
  42. Structural parameterizations for equitable coloring: Complexity, FPT algorithms, and kernelization. Algorithmica, 85:1912–1947, July 2023. doi:10.1007/s00453-022-01085-w.
  43. 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.
  44. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367–375, 2001. doi:10.1006/jcss.2000.1727.
  45. Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size. Journal of Discrete Algorithms, 4(1):142–154, 2006. doi:10.1016/j.jda.2005.01.005.
  46. Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences, 79(1):39–49, 2013. doi:10.1016/j.jcss.2012.04.004.
  47. High-multiplicity N-fold IP via configuration LP. Mathematical Programming, 200(1):199–227, 2023. doi:10.1007/s10107-022-01882-9.
  48. Combinatorial N-fold integer programming and applications. Mathematical Programming, 184(1):1–34, 2020. doi:10.1007/s10107-019-01402-2.
  49. 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. doi:10.1609/aaai.v36i11.21630.
  50. An improved parameterized algorithm for treewidth. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC ’23, pages 528–541. ACM, 2023. doi:10.1145/3564246.3585245.
  51. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19–37, Sep 2012. doi:10.1007/s00453-011-9554-x.
  52. Hendrik W. Lenstra, Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4), 1983. doi:10.1287/moor.8.4.538.
  53. Automated congressional redistricting. ACM Journal of Experimental Algorithmics, 24, April 2019. doi:10.1145/3316513.
  54. 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. doi:10.1007/978-1-4419-7997-1_25.
  55. Most uniform path partitioning and its use in image processing. Discrete Applied Mathematics, 42(2):227–256, 1993. doi:10.1016/0166-218X(93)90048-S.
  56. The parameterised complexity of computing the maximum modularity of a graph. Algorithmica, 82(8):2174–2199, August 2020. doi:10.1007/s00453-019-00649-7.
  57. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity: Graphs, Structures, and Algorithms, volume 28 of Algorithms and Combinatorics. Springer, 2012. doi:10.1007/978-3-642-27875-4.
  58. Rolf Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2006. doi:10.1093/acprof:oso/9780198566076.001.0001.
  59. Manfred Wiegers. The k𝑘kitalic_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.
  60. Justin C. Williams Jr. Political redistricting: A review. Papers in Regional Science, 74(1), 1995. doi:10.1111/j.1435-5597.1995.tb00626.x.

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Paper to Video (Beta)

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Tweets

Sign up for free to view the 2 tweets with 0 likes about this paper.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up for Free