Approximating Submodular k-Partition via Principal Partition Sequence

Authors Karthekeyan Chandrasekaran , Weihang Wang

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Karthekeyan Chandrasekaran
  • University of Illinois, Urbana-Champaign, IL, USA
Weihang Wang
  • University of Illinois, Urbana-Champaign, IL, USA


Karthekeyan Chandrasekaran would like to thank Chandra Chekuri for asking about the approximation factor of the principal partition sequence based approach for symmetric submodular k-partition.

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Karthekeyan Chandrasekaran and Weihang Wang. Approximating Submodular k-Partition via Principal Partition Sequence. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In submodular k-partition, the input is a submodular function f:2^V → ℝ_{≥ 0} (given by an evaluation oracle) along with a positive integer k and the goal is to find a partition of the ground set V into k non-empty parts V_1, V_2, …, V_k in order to minimize ∑_{i=1}^k f(V_i). Narayanan, Roy, and Patkar [Narayanan et al., 1996] designed an algorithm for submodular k-partition based on the principal partition sequence and showed that the approximation factor of their algorithm is 2 for the special case of graph cut functions (which was subsequently rediscovered by Ravi and Sinha [R. Ravi and A. Sinha, 2008]). In this work, we study the approximation factor of their algorithm for three subfamilies of submodular functions - namely monotone, symmetric, and posimodular and show the following results: 1) The approximation factor of their algorithm for monotone submodular k-partition is 4/3. This result improves on the 2-factor that was known to be achievable for monotone submodular k-partition via other algorithms. Moreover, our upper bound of 4/3 matches the recently shown lower bound under polynomial number of function evaluation queries [Santiago, 2021]. Our upper bound of 4/3 is also the first improvement beyond 2 for a certain graph partitioning problem that is a special case of monotone submodular k-partition. 2) The approximation factor of their algorithm for symmetric submodular k-partition is 2. This result generalizes their approximation factor analysis beyond graph cut functions. 3) The approximation factor of their algorithm for posimodular submodular k-partition is 2. We also construct an example to show that the approximation factor of their algorithm for arbitrary submodular functions is Ω(n/k).

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  • Theory of computation → Approximation algorithms analysis
  • Approximation algorithms


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  1. M. Baïou and F. Barahona. Packing Hypertrees and the k-cut problem in Hypergraphs. In International Conference on Learning and Intelligent Optimization, LION, pages 521-534, 2023. Google Scholar
  2. F. Barahona. On the k-cut problem. Operations Research Letters, 26(3):99-105, 2000. Google Scholar
  3. D. Chakrabarty and Z. Huang. Testing coverage functions. In Automata, Languages, and Programming, ICALP, pages 170-181, 2012. Google Scholar
  4. K. Chandrasekaran and C. Chekuri. Hypergraph k-cut for fixed k in deterministic polynomial time. Mathematics of Operations Research, 2022. Prelim. version in FOCS 2020: pages 810-821. Google Scholar
  5. K. Chandrasekaran and W. Wang. Approximating submodular k-partition via principal partition sequence. arXiv, 2023. URL:
  6. C. Chekuri and S. Li. On the hardness of approximating the k-way hypergraph cut problem. Theory Comput., 16:1-8, 2020. Google Scholar
  7. C. Chekuri, K. Quanrud, and C. Xu. LP Relaxation and Tree Packing for Minimum k-Cut. SIAM Journal on Discrete Mathematics, 34(2):1334-1353, 2020. Google Scholar
  8. W. Cunningham. Optimal attack and reinforcement of a network. J. ACM, 32(3):549-561, July 1985. Google Scholar
  9. M. P. Desai, H. Narayanan, and S. B. Patkar. The realization of finite state machines by decomposition and the principal lattice of partitions of a submodular function. Discrete Appl. Math., 131:299-310, 2003. Google Scholar
  10. O. Goldschmidt and D. Hochbaum. A Polynomial Algorithm for the k-cut Problem for Fixed k. Mathematics of Operations Research, 19(1):24-37, February 1994. Google Scholar
  11. V. Kolmogorov. A Faster Algorithm for Computing the Principal Sequence of Partitions of a Graph. Algorithmica, pages 394-412, 2010. Google Scholar
  12. P. Manurangsi. Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis. Algorithms, 11(1):10, 2018. Google Scholar
  13. H. Nagamochi and Y. Kamidoi. Minimum cost subpartitions in graphs. Information Processing Letters, 102(2):79-84, 2007. Google Scholar
  14. K. Nagano, Y. Kawahara, and S. Iwata. Minimum average cost clustering. In Advances in Neural Information Processing Systems, volume 23 of NIPS, 2010. Google Scholar
  15. J. Naor and Y. Rabani. Tree packing and approximating k-cuts. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 26-27, 2001. Google Scholar
  16. H. Narayanan. The principal lattice of partitions of a submodular function. Linear Algebra and its Applications, 144:179-216, 1991. Google Scholar
  17. H. Narayanan. Submodular functions and electrical networks (2nd edition), 1997. URL:
  18. H. Narayanan, S. Roy, and S. Patkar. Approximation algorithms for min-k-overlap problems using the principal lattice of partitions approach. Journal of Algorithms, 21(2):306-330, 1996. Google Scholar
  19. K. Okumoto, T. Fukunaga, and H. Nagamochi. Divide-and-conquer algorithms for partitioning hypergraphs and submodular systems. Algorithmica, 62(3):787-806, 2012. Google Scholar
  20. S. B. Patkar and H. Narayanan. Improving graph partitions using submodular functions. Discrete Appl. Math., 131:535-553, 2003. Google Scholar
  21. K. Quanrud. Fast and Deterministic Approximations for k-Cut. In Proceedings of Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX, pages 23:1-23:20, 2019. Google Scholar
  22. R. Ravi and A. Sinha. Approximating k-cuts using network strength as a lagrangean relaxation. European Journal of Operational Research, 186(1):77-90, 2008. Google Scholar
  23. R. Santiago. New approximations and hardness results for submodular partitioning problems. In Proceedings of International Workshop on Combinatorial Algorithms, IWOCA, pages 516-530, 2021. Google Scholar
  24. H. Saran and V. Vazirani. Finding k Cuts within Twice the Optimal. SIAM Journal on Computing, 24(1):101-108, 1995. Google Scholar
  25. V. Vazirani. Approximation algorithms. Springer, Berlin, Germany, 2003. Google Scholar
  26. L. Zhao, H. Nagamochi, and T. Ibaraki. Greedy splitting algorithms for approximating multiway partition problems. Mathematical Programming, 102(1):167-183, 2005. Google Scholar