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arXiv · 2507.10436

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

Jungho Ahn·Ian DeHaan·Eun Jung Kim·Euiwoong Lee
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Abstract

We present a polynomial-time $(\alpha_{GW} + \varepsilon)$-approximation algorithm for the Maximum Cut problem on interval graphs and split graphs, where $\alpha_{GW} \approx 0.878$ is the approximation guarantee of the Goemans-Williamson algorithm and $\varepsilon > 10^{-34}$ 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, Euiwoong Lee. 2025-07-14. Approximating Maximum Cut on Interval Graphs and Split Graphs beyond Goemans-Williamson. https://arxiv.org/abs/2507.10436

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