Michał Pilipczuk

Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time

For the vast majority of local problems on graphs of small tree width (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c^tw |V|^O(1) time algorithms, where tw is the tree width of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best -- known algorithms were naive dynamic programming schemes running in at least tw^tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c^tw |V|^O(1) time Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Steiner Tree, Feedback Vertex Set and Connected Dominating Set. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and H-minor-free graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like Cycle Packing.

An O(c^k n) 5-Approximation Algorithm for Treewidth

We give an algorithm that for an input n-vertex graph G and integer k > 0, in time O(ckn) either outputs that the tree width of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for tree width which runs in time single-exponential in k and linear in n. Tree width based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single-exponential in the tree width and linear in the input size.

On multiway cut parameterized above lower bounds

In this paper we consider two above lower bound parameterizations of the NodeMultiway Cut problem -- above the maximum separating cut and above a natural LP-relaxation -- and prove them to be fixed-parameter tractable. Our results imply O*(4k) algorithms for Vertex Coverabove Maximum Matching and Almost 2-SAT as well as an O*(2k) algorithm for NodeMultiway Cut with a standard parameterization by the solution size, improving previous bounds for these problems.

Clique Cover and Graph Separation: New Incompressibility Results

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. In this article, we show that, unless the polynomial hierarchy collapses to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter: ---Edge Clique Cover, parameterized by the number of cliques, ---Directed Edge/Vertex Multiway Cut, parameterized by the size of the cutset, even in the case of two terminals, ---Edge/Vertex Multicut, parameterized by the size of the cutset, and ---k-Way Cut, parameterized by the size of the cutset.

A

We give an algorithm that for an input

c^k n

n

5-Approximation Algorithm for Treewidth

-vertex graph

Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask).

G

Tight bounds for parameterized complexity of cluster editing with a small number of clusters

and integer

Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams

k>0