Fast Biclustering by Dual Parameterization

We study two clustering problems, Starforest Editing, the problem of adding and deleting edges to obtain a disjoint union of stars, and the generalization Bicluster Editing. We show that, in addition to being NP-hard, none of the problems can be solved in subexponential time unless the exponential time hypothesis fails. Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the number of connected components in the solution should not make the problem easier: In particular, they argue that the subexponential time algorithm for editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J. Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p is a secondary parameter, bounding the number of components in the solution. However, upon bounding the number of stars or bicliques in the solution, we obtain algorithms which run in time $2^{5 \sqrt{pk}} + O(n+m)$ for p-Starforest Editing and $2^{O(p \sqrt{k} \log(pk))} + O(n+m)$ for p-Bicluster Editing. We obtain a similar result for the more general case of t-Partite p-Cluster Editing. This is subexponential in k for fixed number of clusters, since p is then considered a constant. Our results even out the number of multivariate subexponential time algorithms and give reasons to believe that this area warrants further study.Comment: Accepted for presentation at IPEC 201

Polynomial kernelization for removing induced claws and diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fai

Polynomial Kernelization for Removing Induced Claws and Diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fai

Polynomial kernelization for removing induced claws and diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a K1,3) nor a diamond (a K4 with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph G and a parameter k, the question is whether one can remove at most k edges from G to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time 2o(k)⋅|V(G)|O(1) unless the Exponential Time Hypothesis fai