# Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Benchmark Data

`CVX`package in Matlab for graphs G1 to G21. For graphs G22 to G54, G57 to G59, G62 to G67, and G72, we used Mathematica’s function

`SemidefiniteOptimization`. For graphs G55, G56, G60, G61, G70, G77, and G81 neither

`SemidefiniteOptimization`nor

`CVX`were able to find a solution to the SDP problem. We therefore had to exclude these instances from further consideration. From the SDP solution of each instance, we computed 50 iterations for the randomized clustering algorithms, including Goeamans and Williamson randomized clustering and reported the best solution for the seven best algorithms.

#### 2.2. Clustering in the Goemans–Williamson Algorithm

**Lemma**

**1.**

**Proof.**

- Fuzzy c-means
**(Fuzzy)** - Randomized k-means initialized among vectors ${\overrightarrow{v}}_{i}$
**(K-MeansRand)** - Deterministic k-means
**(K-MeansDet)** - Randomized k-medoids
**(K-MedRand)** - Deterministic k-medoids
**(K-MedDet)** - Minimum Spanning Tree
**(MST)** - Randomized Rounding of Goemans–Williamson
**(RR)** - Randomized k-means initialized with two random vectors
**(K-Means2N)** - Randomized k-means initialized with a random vector and its negative
**(K-MeansNM)**

#### 2.3. Local Search

## 3. Results

#### 3.1. Cluster Quality Correlations with Solution Quality

#### 3.2. An Instance-Specific Approximation Guarantee

#### 3.3. Benchmarking Experiments

#### 3.4. Local Search Improvement

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IQP | Integer Quadratic Programming |

VP | Vector Programming |

SDP | Semidefinite Programming |

Fuzzy | Fuzzy c-means clustering |

K-MeansRand | Randomized version of k-Means |

K-MeansDet | Deterministic version of k-Means |

K-MedRand | Randomized version of k-Medoids |

K-MedDet | Deterministic version of k-Medoids |

MST | Minimum Spanning Tree |

RR | Randomized Rounding (Goemans–Williamson rounding) |

K-Means2N | Randomized version of k-Means initialized with 2 random vectors |

K-MeansNM | Deterministic version of k-Means initialized with a random vector and its negative |

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**Figure 1.**Exploration of the path of 50 iterations of K-MeansNM on the distortion weight of the corresponding cut. The diagram shows all values generated while running the k-means. Points at the bottom right of the plot are the starting points and thus the results of Goemans–Williamson rounding; meanwhile, points at the top left are the end points. At each step of K-MeansNM, the points move to the left until the algorithm finds a local minima of distortion. The red line is a linear fit of all the points after the second step of k-means. These show a clear correlation between cluster distortion and cut weight.

**Figure 2.**Comparison between the cut value found by clustering algorithms and RR, using ${\widehat{f}}_{cluster}$ as the comparison. The horizontal axis represents the graph number, i.e., graph Gi is shown in position i. The red line indicates the average over the benchmark set. Positive values indicate that the clustering solutions are superior to the RR solutions.

**Figure 3.**Comparison of instance-specific performance bounds $\alpha \left(A\right)$ between clustering algorithms and RR. The cyan line is the average of $\alpha \left(A\right)$ for the clustering methods and the magenta line is the average of RR for comparison.

**Figure 4.**Comparison between the cut value found by clustering algorithms and RR after local search (LS), using ${\widehat{f}}_{cluster}$ as the comparison. The horizontal axis represents the graph number, i.e., graph Gi is shown at position i. The red line indicates the average over the benchmark set. Positive values indicate that the locally improved clustering solutions are superior to the locally improved RR solutions.

**Figure 5.**Comparison of the best cut value obtained with clustering (in magenta) and RR (in cyan) and subsequence local improvement with the best cut value (${f}_{best}$) available in the literature.

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**MDPI and ACS Style**

Rodriguez-Fernandez, A.E.; Gonzalez-Torres, B.; Menchaca-Mendez, R.; Stadler, P.F. Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem. *Computation* **2020**, *8*, 75.
https://doi.org/10.3390/computation8030075

**AMA Style**

Rodriguez-Fernandez AE, Gonzalez-Torres B, Menchaca-Mendez R, Stadler PF. Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem. *Computation*. 2020; 8(3):75.
https://doi.org/10.3390/computation8030075

**Chicago/Turabian Style**

Rodriguez-Fernandez, Angel E., Bernardo Gonzalez-Torres, Ricardo Menchaca-Mendez, and Peter F. Stadler. 2020. "Clustering Improves the Goemans–Williamson Approximation for the Max-Cut Problem" *Computation* 8, no. 3: 75.
https://doi.org/10.3390/computation8030075