Document

Approximating Submodular k-Partition via Principal Partition Sequence

File

LIPIcs.APPROX-RANDOM.2023.3.pdf
• Filesize: 0.78 MB
• 16 pages

Acknowledgements

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.

Cite As

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)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.3

Abstract

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).

Subject Classification

ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
Keywords
• Approximation algorithms

Metrics

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

References

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.
2. F. Barahona. On the k-cut problem. Operations Research Letters, 26(3):99-105, 2000.
3. D. Chakrabarty and Z. Huang. Testing coverage functions. In Automata, Languages, and Programming, ICALP, pages 170-181, 2012.
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.
5. K. Chandrasekaran and W. Wang. Approximating submodular k-partition via principal partition sequence. arXiv, 2023. URL: https://arxiv.org/abs/2305.01069.
6. C. Chekuri and S. Li. On the hardness of approximating the k-way hypergraph cut problem. Theory Comput., 16:1-8, 2020.
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.
8. W. Cunningham. Optimal attack and reinforcement of a network. J. ACM, 32(3):549-561, July 1985.
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.
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.
11. V. Kolmogorov. A Faster Algorithm for Computing the Principal Sequence of Partitions of a Graph. Algorithmica, pages 394-412, 2010.
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.
13. H. Nagamochi and Y. Kamidoi. Minimum cost subpartitions in graphs. Information Processing Letters, 102(2):79-84, 2007.
14. K. Nagano, Y. Kawahara, and S. Iwata. Minimum average cost clustering. In Advances in Neural Information Processing Systems, volume 23 of NIPS, 2010.
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.
16. H. Narayanan. The principal lattice of partitions of a submodular function. Linear Algebra and its Applications, 144:179-216, 1991.
17. H. Narayanan. Submodular functions and electrical networks (2nd edition), 1997. URL: https://www.ee.iitb.ac.in/~hn/book/SubmodularFunction_2ed.pdf.
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.
19. K. Okumoto, T. Fukunaga, and H. Nagamochi. Divide-and-conquer algorithms for partitioning hypergraphs and submodular systems. Algorithmica, 62(3):787-806, 2012.
20. S. B. Patkar and H. Narayanan. Improving graph partitions using submodular functions. Discrete Appl. Math., 131:535-553, 2003.
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.
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.
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.
24. H. Saran and V. Vazirani. Finding k Cuts within Twice the Optimal. SIAM Journal on Computing, 24(1):101-108, 1995.
25. V. Vazirani. Approximation algorithms. Springer, Berlin, Germany, 2003.
26. L. Zhao, H. Nagamochi, and T. Ibaraki. Greedy splitting algorithms for approximating multiway partition problems. Mathematical Programming, 102(1):167-183, 2005.