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<inline-formula><math display="inline"><semantics><mrow><mi mathvariant="normal">M</mi><mi mathsize="small">AX</mi><mo>-</mo><mi mathvariant="normal">C</mi><mi mathsize="small">UT</mi></mrow></semantics></math></inline-formula> is one of the well-studied NP-hard combinatorial optimization problems. It can be formulated as an Integer Quadratic Programming problem and admits a simple relaxation obtained by replacing the integer “spin” variables <inline-formula><math display="inline"><semantics><msub><mi>x</mi><mi>i</mi></msub></semantics></math></inline-formula> by unitary vectors <inline-formula><math display="inline"><semantics><msub><mover accent="true"><mi>v</mi><mo stretchy="false">→</mo></mover><mi>i</mi></msub></semantics></math></inline-formula>. The Goemans–Williamson rounding algorithm assigns the solution vectors of the relaxed quadratic program to a corresponding integer spin depending on the sign of the scalar product <inline-formula><math display="inline"><semantics><mrow><msub><mover accent="true"><mi>v</mi><mo stretchy="false">→</mo></mover><mi>i</mi></msub><mo>·</mo><mover accent="true"><mi>r</mi><mo stretchy="false">→</mo></mover></mrow></semantics></math></inline-formula> with a random vector <inline-formula><math display="inline"><semantics><mover accent="true"><mi>r</mi><mo stretchy="false">→</mo></mover></semantics></math></inline-formula>. Here, we investigate whether better graph cuts can be obtained by instead using a more sophisticated clustering algorithm. We answer this question affirmatively. Different initializations of <i>k</i>-means and <i>k</i>-medoids clustering produce better cuts for the graph instances of the most well known benchmark for <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="normal">M</mi><mi mathsize="small">AX</mi><mo>-</mo><mi mathvariant="normal">C</mi><mi mathsize="small">UT</mi></mrow></semantics></math></inline-formula>. In particular, we found a strong correlation of cluster quality and cut weights during the evolution of the clustering algorithms. Finally, since in general the maximal cut weight of a graph is not known beforehand, we derived instance-specific lower bounds for the approximation ratio, which give information of how close a solution is to the global optima for a particular instance. For the graphs in our benchmark, the instance specific lower bounds significantly exceed the Goemans–Williamson guarantee.