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Approximating Maximum Cut on Interval Graphs and Split Graphs Beyond Goemans-Williamson

Authors Jungho Ahn , Ian DeHaan , Eun Jung Kim , Euiwoong Lee

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  • Theory of computation → Approximation algorithms analysis
Keywords
  • Maximum cut
  • graph theory
  • interval graphs
  • split graphs

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Abstract

We present a polynomial-time (α_{GW} + ε)-approximation algorithm for the Maximum Cut problem on interval graphs and split graphs, where α_{GW} ≈ 0.878 is the approximation guarantee of the Goemans-Williamson algorithm and ε > 10^{-34} is a fixed constant. To attain this, we give an improved analysis of a slight modification of the Goemans-Williamson algorithm for graphs in which triangles can be packed into a constant fraction of their edges. We then pair this analysis with structural results showing that both interval graphs and split graphs either have such a triangle packing or have maximum cut close to their number of edges. We also show that, subject to the Small Set Expansion Hypothesis, there exists a constant c > 0 such that there is no polyomial-time (1 - c)-approximation for Maximum Cut on split graphs.

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Jungho Ahn, Ian DeHaan, Eun Jung Kim, and Euiwoong Lee. Approximating Maximum Cut on Interval Graphs and Split Graphs Beyond Goemans-Williamson. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 20:1-20:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.20

Author Details

Jungho Ahn
  • Inha University, Incheon, South Korea
Ian DeHaan
  • University of Michigan, Ann Arbor, MI, USA
Eun Jung Kim
  • KAIST and IBS, Daejon, South Korea
  • CNRS, Paris, France
Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Ahn, Jungho: Supported by Leverhulme Trust Research Project Grant RPG-2024-182.
  • Kim, Eun Jung: Supported by Institute for Basic Science (IBS-R029-C1) and National Research Foundation, Korea (RS-2025-00563533).
  • Lee, Euiwoong: Supported in part by NSF grant CCF-2236669.

References

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