𝓁_p-Norm Multiway Cut

Authors Karthekeyan Chandrasekaran , Weihang Wang

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

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Karthekeyan Chandrasekaran and Weihang Wang. 𝓁_p-Norm Multiway Cut. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We introduce and study 𝓁_p-norm-multiway-cut: the input here is an undirected graph with non-negative edge weights along with k terminals and the goal is to find a partition of the vertex set into k parts each containing exactly one terminal so as to minimize the 𝓁_p-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when p = 1) and min-max multiway cut (when p = ∞), both of which are well-studied classic problems in the graph partitioning literature. We show that 𝓁_p-norm-multiway-cut is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an O(log² n)-approximation for all p ≥ 1. We also show an integrality gap of Ω(k^{1-1/p}) for a natural convex program and an O(k^{1-1/p-ε})-inapproximability for any constant ε > 0 assuming the small set expansion hypothesis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • multiway cut
  • approximation algorithms


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