Subexponential algorithm for-cluster edge deletion: Exception or rule?

Neeldhara Misra, Fahad Panolan, Saket Saurabh

Abstract: We study the question of finding a set of at most k edges, whose removal makes the input n-vertex graph a disjoint union of s-clubs (graphs of diameter s). Komusiewicz and Uhlmann [DAM 2012] showed that Cluster Edge Deletion (i.e., for the case of 1-clubs (cliques)), cannot be solved in time 2o(k)nO(1)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">o</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> unless the Exponential Time Hypothesis (ETH) fails. But, Fomin et al. [JCSS 2014] showed that if the number of cliques in the output graph is restricted to d, then the problem (d-Cluster Edge Deletion) can be solved in time O(2O(dk)+m+n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msqrt is="true"><mrow is="true"><mi is="true">d</mi><mi is="true">k</mi></mrow></msqrt><mo stretchy="false" is="true">)</mo></mrow></msup><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">m</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math>. We show that assuming ETH, there is no algorithm solving 2-Club Cluster Edge Deletion in time 2o(k)nO(1)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">o</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math>. Further, we show that the same lower bound holds in the case of s-Club d-Cluster Edge Deletion for any s≥2<math><mi is="true">s</mi><mo is="true">≥</mo><mn is="true">2</mn></math> and d≥2<math><mi is="true">d</mi><mo is="true">≥</mo><mn is="true">2</mn></math>.