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Approximation Algorithm for Norm Multiway Cut

Authors Charlie Carlson, Jafar Jafarov, Konstantin Makarychev, Yury Makarychev, Liren Shan

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Author Details

Charlie Carlson
  • University of Colorado Boulder, CO, USA
Jafar Jafarov
  • Toyota Technological Institute at Chicago, IL, USA
Konstantin Makarychev
  • Northwestern University, Evanston, IL, USA
Yury Makarychev
  • Toyota Technological Institute at Chicago, IL, USA
Liren Shan
  • Northwestern University, Evanston, IL, USA

Funding

  • Makarychev, Konstantin: supported by NSF Awards CCF-1955351, CCF-1934931, EECS-2216970.
  • Makarychev, Yury: supported by NSF CCF-1955173, CCF-1934843, and ECCS-2216899.
  • Shan, Liren: supported by NSF Awards CCF-1955351, CCF-1934931, and EECS-2216970.

Cite AsGet BibTex

Charlie Carlson, Jafar Jafarov, Konstantin Makarychev, Yury Makarychev, and Liren Shan. Approximation Algorithm for Norm Multiway Cut. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.32

Abstract

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph G into k parts so as to separate k given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced 𝓁_p-norm Multiway Cut, a generalization of the problem, in which the goal is to minimize the 𝓁_p norm of the edge boundaries of k parts. We provide an O(log^{1/2} nlog^{1/2+1/p} k) approximation algorithm for this problem, improving upon the approximation guarantee of O(log^{3/2} n log^{1/2} k) due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an O(log^{1/2} n log^{7/2} k) approximation algorithm with a weaker oracle and an O(log^{1/2} n log^{5/2} k) approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no n^{1/4-ε} approximation algorithm for every ε > 0 assuming the Hypergraph Dense-vs-Random Conjecture.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
Keywords
  • Multiway cut
  • Approximation algorithms

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References

  1. Saba Ahmadi, Samir Khuller, and Barna Saha. Min-max correlation clustering via multicut. In Proceedings of the Integer Programming and Combinatorial Optimization, pages 13-26, 2019. Google Scholar
  2. Haris Angelidakis, Yury Makarychev, and Pasin Manurangsi. An improved integrality gap for the Călinescu-Karloff-Rabani relaxation for multiway cut. In Proceedings of Integer Programming and Combinatorial Optimization, pages 39-50. Springer, 2017. Google Scholar
  3. Nikhil Bansal, Uriel Feige, Robert Krauthgamer, Konstantin Makarychev, Viswanath Nagarajan, Joseph Seffi, and Roy Schwartz. Min-max graph partitioning and small set expansion. SIAM Journal on Computing, 43(2):872-904, 2014. Google Scholar
  4. Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Vivek Madan. Improving the integrality gap for multiway cut. Mathematical Programming, 183(1-2):171-193, 2020. Google Scholar
  5. Niv Buchbinder, Joseph Naor, and Roy Schwartz. Simplex partitioning via exponential clocks and the multiway cut problem. In Proceedings of the Symposium on Theory of Computing, pages 535-544, 2013. Google Scholar
  6. Niv Buchbinder, Roy Schwartz, and Baruch Weizman. Simplex transformations and the multiway cut problem. In Proceedings of the Symposium on Discrete Algorithms, pages 2400-2410, 2017. Google Scholar
  7. Gruia Călinescu, Howard Karloff, and Yuval Rabani. An improved approximation algorithm for multiway cut. In Proceedings of the Symposium on Theory of Computing, pages 48-52, 1998. Google Scholar
  8. Karthekeyan Chandrasekaran and Weihang Wang. 𝓁_p-norm multiway cut. Algorithmica, 84(9):2667-2701, 2022. Google Scholar
  9. Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. Google Scholar
  10. Ari Freund and Howard Karloff. A lower bound of 8/(7+ 1/(k- 1)) on the integrality ratio of the Călinescu-Karloff-Rabani relaxation for multiway cut. Information Processing Letters, 75(1-2):43-50, 2000. Google Scholar
  11. David R Karger, Philip Klein, Cliff Stein, Mikkel Thorup, and Neal E Young. Rounding algorithms for a geometric embedding of minimum multiway cut. In Proceedings of the Symposium on Theory of Computing, pages 668-678, 1999. Google Scholar
  12. Rajsekar Manokaran, Joseph Naor, Prasad Raghavendra, and Roy Schwartz. SDP gaps and UGC hardness for multiway cut, 0-extension, and metric labeling. In Proceedings of the Symposium on Theory of Computing, pages 11-20, 2008. Google Scholar
  13. Ankit Sharma and Jan Vondrák. Multiway cut, pairwise realizable distributions, and descending thresholds. In Proceedings of the Symposium on Theory of Computing, pages 724-733, 2014. Google Scholar
  14. Zoya Svitkina and Éva Tardos. Min-max multiway cut. In Proceedings of Approximation, Randomization, and Combinatorial Optimization, pages 207-218, 2004. Google Scholar
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