# Complex Semidefinite Programming and Max-k-Cut

## TL;DR

This paper extends the semidefinite programming approach for Max-$k$-Cut, providing a simple rounding algorithm that achieves near state-of-the-art approximation ratios for general $k$, building on and generalizing prior work for Max-3-Cut.

## Contribution

It introduces a new rounding algorithm for Max-$k$-Cut that generalizes Goemans and Williamson's method, achieving approximation guarantees close to the best known results.

## Key findings

- The new rounding algorithm matches Goemans and Williamson's analysis for Max-3-Cut.
- For $k \\geq 4$, the approximation ratios are about 0.01 worse than the best known.
- A generalized version of the algorithm potentially matches the best guarantees of De Klerk et al.

## Abstract

In a second seminal paper on the application of semidefinite programming to graph partitioning problems, Goemans and Williamson showed how to formulate and round a complex semidefinite program to give what is to date still the best-known approximation guarantee of .836008 for Max-$3$-Cut. (This approximation ratio was also achieved independently by De Klerk et al.) Goemans and Williamson left open the problem of how to apply their techniques to Max-$k$-Cut for general $k$. They point out that it does not seem straightforward or even possible to formulate a good quality complex semidefinite program for the general Max-$k$-Cut problem, which presents a barrier for the further application of their techniques. We present a simple rounding algorithm for the standard semidefinite programmming relaxation of Max-$k$-Cut and show that it is equivalent to the rounding of Goemans and Williamson in the case of Max-$3$-Cut. This allows us to transfer the elegant analysis of Goemans and Williamson for Max-3-Cut to Max-$k$-Cut. For $k \geq 4$, the resulting approximation ratios are about $.01$ worse than the best known guarantees. Finally, we present a generalization of our rounding algorithm and conjecture (based on computational observations) that it matches the best-known guarantees of De Klerk et al.