On the Approximability of Max-Cut on 3-Colorable Graphs and Graphs with Large Independent Sets
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918440816607232 |
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| author | Ghoshal, Suprovat Huang, Neng Lee, Euiwoong Makarychev, Konstantin Makarychev, Yury |
| author_facet | Ghoshal, Suprovat Huang, Neng Lee, Euiwoong Makarychev, Konstantin Makarychev, Yury |
| contents | Max-Cut is a classical graph-partitioning problem where given a graph $G = (V,E)$, the objective is to find a cut $(S,S^c)$ which maximizes the number of edges crossing the cut. In a seminal work, Goemans and Williamson gave an $α_{GW} \approx 0.87856$-factor approximation algorithm for the problem, which was later shown to be tight by the work of Khot, Kindler, Mossel, and O'Donnell. Since then, there has been a steady progress in understanding the approximability at even finer levels, and a fundamental goal in this context is to understand how the structure of the underlying graph affects the approximability of the Max-Cut problem.
In this work, we investigate this question by exploring how the chromatic structure of a graph affects the Max-Cut problem. While it is well-known that Max-Cut can be solved perfectly and near-perfectly in $2$-colorable and almost $2$-colorable graphs in polynomial time, here we explore its approximability under much weaker structural conditions such as when the graph is $3$-colorable or contains a large independent set. Our main contributions in this context are as follows:
1. We show Max-Cut is $α_{GW}$-hard to approximate for $3$-colorable graphs.
2. We identify a natural threshold $α^*$ such that the following holds. Firstly, for graphs which contain an independent set of size up to $α^*$, Max-Cut continues to be $α_{GW}$-factor hard to approximate. Furthermore, for any graph that contains an independent set of size $> α^*$, there exists an efficient $>α_{GW}$-approximation algorithm for Max-Cut.
Our hardness results are derived using various analytical tools and novel variants of the Majority-Is-Stablest theorem, which might be of independent interest. Our algorithmic results are based on a novel SDP relaxation, which is then rounded and analyzed using interval arithmetic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10318 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Approximability of Max-Cut on 3-Colorable Graphs and Graphs with Large Independent Sets Ghoshal, Suprovat Huang, Neng Lee, Euiwoong Makarychev, Konstantin Makarychev, Yury Data Structures and Algorithms Max-Cut is a classical graph-partitioning problem where given a graph $G = (V,E)$, the objective is to find a cut $(S,S^c)$ which maximizes the number of edges crossing the cut. In a seminal work, Goemans and Williamson gave an $α_{GW} \approx 0.87856$-factor approximation algorithm for the problem, which was later shown to be tight by the work of Khot, Kindler, Mossel, and O'Donnell. Since then, there has been a steady progress in understanding the approximability at even finer levels, and a fundamental goal in this context is to understand how the structure of the underlying graph affects the approximability of the Max-Cut problem. In this work, we investigate this question by exploring how the chromatic structure of a graph affects the Max-Cut problem. While it is well-known that Max-Cut can be solved perfectly and near-perfectly in $2$-colorable and almost $2$-colorable graphs in polynomial time, here we explore its approximability under much weaker structural conditions such as when the graph is $3$-colorable or contains a large independent set. Our main contributions in this context are as follows: 1. We show Max-Cut is $α_{GW}$-hard to approximate for $3$-colorable graphs. 2. We identify a natural threshold $α^*$ such that the following holds. Firstly, for graphs which contain an independent set of size up to $α^*$, Max-Cut continues to be $α_{GW}$-factor hard to approximate. Furthermore, for any graph that contains an independent set of size $> α^*$, there exists an efficient $>α_{GW}$-approximation algorithm for Max-Cut. Our hardness results are derived using various analytical tools and novel variants of the Majority-Is-Stablest theorem, which might be of independent interest. Our algorithmic results are based on a novel SDP relaxation, which is then rounded and analyzed using interval arithmetic. |
| title | On the Approximability of Max-Cut on 3-Colorable Graphs and Graphs with Large Independent Sets |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.10318 |