MAX−CUT 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 xi by unitary vectors v⃗ i. 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 v⃗ i⋅r⃗ with a random vector r⃗ . 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 k-means and k-medoids clustering produce better cuts for the graph instances of the most well known benchmark for MAX−CUT. 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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:84696 |
| Date | 13 April 2023 |
| Creators | Rodriguez-Fernandez, Angel E., Gonzalez-Torres, Bernardo, Menchaca-Mendez, Ricardo, Stadler, Peter F. |
| Publisher | MDPI |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:article, info:eu-repo/semantics/article, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | 2079-3197, 75 |
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