Balanced Crown Decomposition for Connectivity Constraints

Authors Katrin Casel , Tobias Friedrich , Davis Issac , Aikaterini Niklanovits , Ziena Zeif

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Author Details

Katrin Casel
  • Hasso Plattner Institute, University of Potsdam, Germany
Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
Davis Issac
  • Hasso Plattner Institute, University of Potsdam, Germany
Aikaterini Niklanovits
  • Hasso Plattner Institute, University of Potsdam, Germany
Ziena Zeif
  • Hasso Plattner Institute, University of Potsdam, Germany

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Katrin Casel, Tobias Friedrich, Davis Issac, Aikaterini Niklanovits, and Ziena Zeif. Balanced Crown Decomposition for Connectivity Constraints. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • crown decomposition
  • connected partition
  • balanced partition
  • approximation algorithms


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  1. Ravindra K Ahuja, Andrew V Goldberg, James B Orlin, and Robert E Tarjan. Finding minimum-cost flows by double scaling. Mathematical programming, 53(1-3):243-266, 1992. Google Scholar
  2. Curtis A Barefoot, Roger Entringer, and Henda Swart. Vulnerability in graphs a comparative survey. Journal of Combinatorial Mathematics and Combinatorial Computing, 1:13-22, 1998. Google Scholar
  3. Ralf Borndörfer, Ziena Elijazyfer, and Stephan Schwartz. Approximating balanced graph partitions. Technical Report 19-25, ZIB, Takustr. 7, 14195 Berlin, 2019. Google Scholar
  4. Christoph Brause and Ingo Schiermeyer. Kernelization of the 3-path vertex cover problem. Discrete Mathematics, 339(7):1935-1939, 2016. URL:
  5. Paolo M Camerini, Giulia Galbiati, and Francesco Maffioli. On the complexity of finding multi-constrained spanning trees. Discrete Applied Mathematics, 5(1):39-50, 1983. Google Scholar
  6. L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac. Spanning tree congestion and computation of generalized györi-lovász partition. In 45th International Colloquium on Automata, Languages, and Programming, volume 107 of LIPIcs, pages 32:1-32:14, 2018. Google Scholar
  7. Frédéric Chataigner, Liliane Benning Salgado, and Yoshiko Wakabayashi. Approximation and inapproximability results on balanced connected partitions of graphs. Discrete Mathematics and Theoretical Computer Science, 9(1), 2007. URL:
  8. Guangting Chen, Yong Chen, Zhi-Zhong Chen, Guohui Lin, Tian Liu, and An Zhang. Approximation algorithms for the maximally balanced connected graph tripartition problem. Journal of Combinatorial Optimization, pages 1-21, 2020. Google Scholar
  9. Jianer Chen, Henning Fernau, Peter Shaw, Jianxin Wang, and Zhibiao Yang. Kernels for packing and covering problems. In Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, pages 199-211. Springer, 2012. Google Scholar
  10. Jiangzhuo Chen, Robert D Kleinberg, László Lovász, Rajmohan Rajaraman, Ravi Sundaram, and Adrian Vetta. (almost) tight bounds and existence theorems for single-commodity confluent flows. Journal of the ACM (JACM), 54(4):16, 2007. Google Scholar
  11. Yong Chen, Zhi-Zhong Chen, Guohui Lin, Yao Xu, and An Zhang. Approximation algorithms for maximally balanced connected graph partition. In International Conference on Combinatorial Optimization and Applications, pages 130-141. Springer, 2019. Google Scholar
  12. Janka Chlebíková. Approximating the maximally balanced connected partition problem in graphs. Information Processing Letters, 60(5):225-230, 1996. Google Scholar
  13. Benny Chor, Mike Fellows, and David W. Juedes. Linear kernels in linear time, or how to save k colors in o(n^2) steps. In Graph-Theoretic Concepts in Computer Science, 30th International Workshop, volume 3353 of LNCS, pages 257-269. Springer, 2004. Google Scholar
  14. Pål Grønås Drange, Markus S. Dregi, and Pim van 't Hof. On the computational complexity of vertex integrity and component order connectivity. Algorithmica, 76(4):1181-1202, 2016. URL:
  15. Rosa Enciso, Michael R Fellows, Jiong Guo, Iyad Kanj, Frances Rosamond, and Ondřej Suchy. What makes equitable connected partition easy. In International Workshop on Parameterized and Exact Computation, pages 122-133. Springer, 2009. Google Scholar
  16. Shimon Even and Robert Endre Tarjan. Computing an st-numbering. Theoretical Computer Science, 2(3):339-344, 1976. Google Scholar
  17. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. SIAM Journal on Discrete Mathematics, 30(1):383-410, 2016. Google Scholar
  18. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  19. Greg N. Frederickson. Optimal algorithms for tree partitioning. In Proceedings of the Second Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, pages 168-177. ACM/SIAM, 1991. URL:
  20. E Gyori. On division of graphs to connected subgraphs, combinatorics. In Colloq. Math. Soc. Janos Bolyai, 1976, 1976. Google Scholar
  21. Alexander Hoyer. On the Independent Spanning Tree Conjectures and Related Problems. PhD thesis, Georgia Institute of Technology, 2019. Google Scholar
  22. Jörg Kalcsics and Roger Z. Ríos-Mercado. Districting Problems, pages 705-743. Springer International Publishing, Cham, 2019. Google Scholar
  23. Atsushi Kaneko, Alexander K. Kelmans, and Tsuyoshi Nishimura. On packing 3-vertex paths in a graph. Journal of Graph Theory, 36(4):175-197, 2001. URL:;2-T.
  24. Mithilesh Kumar and Daniel Lokshtanov. A 2lk kernel for l-component order connectivity. In 11th International Symposium on Parameterized and Exact Computation, volume 63 of LIPIcs, pages 20:1-20:14, 2016. URL:
  25. Sukhamay Kundu and Jayadev Misra. A linear tree partitioning algorithm. SIAM Journal on Computing, 6(1):151-154, 1977. Google Scholar
  26. Wenjun Li, Junjie Ye, and Yixin Cao. Kernelization for p_2-packing: A gerrymandering approach. In Frontiers in Algorithmics - 12th International Workshop, volume 10823 of LNCS, pages 140-153. Springer, 2018. URL:
  27. Richard J. Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM Journal of Applied Mathematics, 36:177–-189, 1979. Google Scholar
  28. László Lovász. A homology theory for spanning tress of a graph. Acta Mathematica Academiae Scientiarum Hungarica, 30(3-4):241-251, 1977. Google Scholar
  29. James B Orlin. Max flows in o(nm) time, or better. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 765-774, 2013. Google Scholar
  30. Yehoshua Perl and Stephen R Schach. Max-min tree partitioning. Journal of the ACM (JACM), 28(1):5-15, 1981. Google Scholar
  31. Arnold L. Rosenberg and Lenwood S. Heath. Graph separators with applications. Frontiers of computer science. Springer, 2001. Google Scholar
  32. Hitoshi Suzuki, Naomi Takahashi, and Takao Nishizeki. A linear algorithm for bipartition of biconnected graphs. Information Processing Letters, 33(5):227-231, 1990. Google Scholar
  33. Robert Endre Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146-160, 1972. Google Scholar
  34. Jianhua Tu, Lidong Wu, Jing Yuan, and Lei Cui. On the vertex cover p_3 problem parameterized by treewidth. Journal of Combinatorial Optimization, 34(2):414-425, 2017. URL:
  35. Koichi Wada and Kimio Kawaguchi. Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 132-143. Springer, 1993. Google Scholar
  36. Jianxin Wang, Dan Ning, Qilong Feng, and Jianer Chen. An improved kernelization for p_2-packing. Information Processing Letters, 110(5):188-192, 2010. URL:
  37. Lele Wang, Zhao Zhang, Di Wu, Weili Wu, and Lidan Fan. Max-min weight balanced connected partition. Journal of Global Optimization, 57(4):1263-1275, 2013. Google Scholar
  38. Bang Ye Wu. A 7/6-approximation algorithm for the max-min connected bipartition problem on grid graphs. In International Conference on Computational Geometry, Graphs and Applications, pages 188-194. Springer, 2010. Google Scholar
  39. Bang Ye Wu. Fully polynomial-time approximation schemes for the max-min connected partition problem on interval graphs. Discrete Mathematics, Algorithms and Applications, 4(1), 2012. Google Scholar
  40. Mingyu Xiao. Linear kernels for separating a graph into components of bounded size. Journal of Computer and System Sciences, 88:260-270, 2017. Google Scholar
  41. Mingyu Xiao and Shaowei Kou. Kernelization and parameterized algorithms for 3-path vertex cover. In Theory and Applications of Models of Computation, volume 10185 of LNCS, pages 654-668, 2017. URL:
  42. Mihalis Yannakakis. Node-deletion problems on bipartite graphs. SIAM Journal on Computing, 10(2):310-327, 1981. URL:
  43. Xing Zhou, Huaimin Wang, Bo Ding, Tianjiang Hu, and Suning Shang. Balanced connected task allocations for multi-robot systems: An exact flow-based integer program and an approximate tree-based genetic algorithm. Expert Systems with Applications, 116:10-20, 2019. Google Scholar
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