On the parameterized complexity of consensus clustering

Martin Dörnfelder, Jiong Guo, Christian Komusiewicz, Mathias Weller

Abstract: Given a collection C<math><mi mathvariant="script" is="true">C</mi></math> of partitions of a base set S, the NP-hard Consensus Clustering problem asks for a partition of S which has a total Mirkin distance of at most t to the partitions in C<math><mi mathvariant="script" is="true">C</mi></math>, where t is a nonnegative integer. We present a parameterized algorithm for Consensus Clustering with running time O(4.24k⋅k3+|C|⋅|S|2)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">4.24</mn></mrow><mrow is="true"><mi is="true">k</mi></mrow></msup><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup><mo is="true">+</mo><mo stretchy="false" is="true">|</mo><mi mathvariant="script" is="true">C</mi><mo stretchy="false" is="true">|</mo><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo><mi is="true">S</mi><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math>, where k:=t/|C|<math><mi is="true">k</mi><mo is="true">:</mo><mo is="true">=</mo><mi is="true">t</mi><mo stretchy="false" is="true">/</mo><mo stretchy="false" is="true">|</mo><mi mathvariant="script" is="true">C</mi><mo stretchy="false" is="true">|</mo></math> is the average Mirkin distance of the solution partition to the partitions of C<math><mi mathvariant="script" is="true">C</mi></math>. Furthermore, we strengthen previous hardness results for Consensus Clustering, showing that Consensus Clustering remains NP-hard even when all input partitions contain at most two subsets. Finally, we study a local search variant of Consensus Clustering, showing W[1]-hardness for the parameter “radius of the Mirkin-distance neighborhood”. In the process, we also consider a local search variant of the related Cluster Editing problem, showing W[1]-hardness for the parameter “radius of the edge modification neighborhood”.