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

• Main Authors: Angel E. Rodriguez-Fernandez, Bernardo Gonzalez-Torres, Ricardo Menchaca-Mendez, and Peter F. Stadler
• Format: Article
• Language: English
• Published: MDPI AG 2020-08-01
• Series: Computation
• Subjects: algorithms; approximation; semidefinite programming; Max-Cut; clustering
• Online Access: https://www.mdpi.com/2079-3197/8/3/75

MAXmathsizesmall-CUTmathsizesmall 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>→</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>→</mo></mover><mi>i</mi></msub><mo>·</mo><mover accent="true"><mi>r</mi><mo>→</mo></mover></mrow></semantics></math></inline-formula> with a random vector <inline-formula><math display="inline"><semantics><mover accent="true"><mi>r</mi><mo>→</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 MAXmathsizesmall-CUTmathsizesmall. 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.