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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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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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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.

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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

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.

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