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)$2^{o(k)}{n}^{O(1)}$ 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)$O(2^{O(\sqrt{dk})}+m+n)$. We show that assuming ETH, there is no algorithm solving 2-Club Cluster Edge Deletion in time 2o(k)nO(1)$2^{o(k)}{n}^{O(1)}$. Further, we show that the same lower bound holds in the case of s-Club d-Cluster Edge Deletion for any s≥2$s\ge 2$ and d≥2$d\ge 2$.