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

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 O(2^{3*sqrt(pk)} + n + m) for p-Starforest Editing and O(2^{O(p * sqrt(k) * log(pk))} + 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 a 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

Definable inapproximability: New challenges for duplicator

AbstractWe consider the hardness of approximation of optimization problems from the point of view of definability. For many $\textrm{NP}$-hard optimization problems it is known that, unless $\textrm{P} = \textrm{NP}$, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.</jats:p