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# Multicriteria Cuts and Size-Constrained k-Cuts in Hypergraphs

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## Cite As

Calvin Beideman, Karthekeyan Chandrasekaran, and Chao Xu. Multicriteria Cuts and Size-Constrained k-Cuts in Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.17

## Abstract

We address counting and optimization variants of multicriteria global min-cut and size-constrained min-k-cut in hypergraphs. 1) For an r-rank n-vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O(r2^{tr}n^{3t-1}). In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne [Aissi et al., 2015]. In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 2) We also address node-budgeted multiobjective min-cuts: For an n-vertex hypergraph endowed with t vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is O(r2^{r}n^{t+2}), where r is the rank of the hypergraph, and the number of node-budgeted b-multiobjective min-cuts for a fixed budget-vector b ∈ ℝ^t_+ is O(n²). 3) We show that min-k-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant k, thus resolving an open problem posed by Queyranne [Guinez and Queyranne, 2012]. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger [Karger, 1993]. Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained k-cuts in hypergraphs.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Combinatorial optimization
##### Keywords
• Multiobjective Optimization
• Hypergraph min-cut
• Hypergraph-k-cut

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## References

1. H. Aissi, A. Mahjoub, T. McCormick, and M. Queyranne. Strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs. Mathematical Programming (Preliminary version in IPCO 2014), 154(1-2):3-28, 2015.
2. H. Aissi, A. Mahjoub, and R. Ravi. Randomized Contractions for Multiobjective Minimum Cuts. In Proceedings of the 25th Annual European Symposium on Algorithms, ESA, pages 6:1-6:13, 2017.
3. A. Armon and U. Zwick. Multicriteria global minimum cuts. Algorithmica, 46(1):15-26, 2006.
4. C. Beideman, K. Chandrasekaran, and C. Xu. Multicriteria Cuts and Size-Constraiend k-Cuts. arXiv, 2020. URL: http://arxiv.org/abs/2006.11589.
5. K. Chandrasekaran, C. Xu, and X. Yu. Hypergraph k-cut in randomized polynomial time. Mathematical Programming (Preliminary version in SODA 2018), November 2019.
6. C. Chekuri and C. Xu. Minimum cuts and sparsification in hypergraphs. SIAM Journal on Computing, 47(6):2118-2156, 2018.
7. E. Dinits, A. Karzanov, and M. Lomonosov. On the structure of a family of minimal weighted cuts in a graph. Studies in Discrete Optimizatoin (in Russian), pages 290-306, 1976.
8. M. Ghaffari, D. Karger, and D. Panigrahi. Random contractions and sampling for hypergraph and hedge connectivity. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1101–-1114, 2017.
9. M. Goemans and J. Soto. Algorithms for symmetric submodular function minimization under hereditary constraints and generalizations. SIAM Journal on Discrete Mathematics, 27(2):1123-1145, 2013.
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. F. Guinez and M. Queyranne. The size-constrained submodular k-partition problem. Unpublished manuscript. Available at https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxmbGF2aW9ndWluZXpob21lcGFnZXxneDo0NDVlMThkMDg4ZWRlOGI1. See also https://smartech.gatech.edu/bitstream/handle/1853/43309/Queyranne.pdf, 2012.
12. D. Karger. Global Min-Cuts in RNC, and Other Ramifications of a Simple Min-Cut Algorithm. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, page 21–30, 1993.
13. D. Karger. Enumerating parametric global minimum cuts by random interleaving. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing, STOC, pages 542-555, 2016.
14. D. Kogan and R. Krauthgamer. Sketching cuts in graphs and hypergraphs. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS, pages 367-376, 2015.
15. K. Mulmuley. Lower bounds in a parallel model without bit operations. SIAM Journal on Computing, 28(4):1460-1509, 1999.
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