Marcin 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.

Exact and approximate bandwidth

In this paper we gather several improvements in the field of exact and approximate exponential time algorithms for the Bandwidth problem. For graphs with treewidth t we present an O(n^O^(^t^)2^n) exact algorithm. Moreover, for any two positive integers k>=2,r>=1, we present a (2kr-1)-approximation algorithm that solves Bandwidth for an arbitrary input graph in O^*(k^n^(^k^-^1^)^r) time and polynomial space where by O^* we denote the standard big O notation but omitting polynomial factors. Finally, we improve the currently best known exact algorithm for arbitrary graphs with an O(4.383^n) time and space algorithm. In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Bjorklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O^*(2^n) time and space, what matches the best known results.

Faster deterministic Feedback Vertex Set

We present a new deterministic algorithm for the Feedback Vertex Set problem parameterized by the solution size. Our algorithm runs in O^@?((2+@f)^k) time, where @f<1.619 is the golden ratio, surpassing the previously fastest O^@?((1+22)^k)-time deterministic algorithm due to Cao et al. (2010) [6]. In our development we follow the approach of Cao et al.; however, thanks to a new reduction rule, we obtain not only better dependency on the parameter in the running time, but also a solution with simple analysis and only a single branching rule.

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.

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

In the Cluster Editing problem, also known as Correlation Clustering, we are given an undirected n-vertex graph G and a positive integer k. The task is to decide if G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, i.e. by adding/deleting at most k edges. We give a subexponential-time parameterized algorithm that in time View the MathML source decides whether G can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies. Our algorithmic findings are complemented by the following tight lower bound on the asymptotic behavior of our algorithm. We show that unless ETH fails, for any constant 0<σ≤1, there is p=Θ(kσ) such that there is no algorithm deciding in time View the MathML source whether G can be transformed into a cluster graph with at most p cliques by changing at most k adjacencies.

Faster Exact Bandwidth

We deal with exact algorithms for Bandwidth , a long studied NP-hard problem. For a long time nothing better than the trivial O *(n !) exhaustive search was known. In 2000, Feige an Kilian [4] came up with a O *(10 n )-time algorithm. Since then there has been a growing interest in exponential time algorithms but this bound has not been improved. In this paper we present a new and quite simple O *(5 n ) algorithm. We also obtain even better bound in some special cases.

Tight bounds for parameterized complexity of Cluster Editing

In the Correlation Clustering problem, also known as Cluster Editing, we are given an undirected graph G and a positive integer k; the task is to decide whether G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, that is, by adding or deleting at most k edges. The motivation of the problem stems from various tasks in computational biology (Ben-Dor et al., Journal of Computational Biology 1999) and machine learning (Bansal et al., Machine Learning 2004). Although in general Correlation Clustering is APX-hard (Charikar et al., FOCS 2003), the version of the problem where the number of cliques may not exceed a prescribed constant p admits a PTAS (Giotis and Guruswami, SODA 2006). We study the parameterized complexity of Correlation Clustering with this restriction on the number of cliques to be created. We give an algorithm that - in time O(2^{O(sqrt{pk})} + n+m) decides whether a graph G on n vertices and m edges can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies. We complement these algorithmic findings by the following, surprisingly tight lower bound on the asymptotic behavior of our algorithm. We show that unless the Exponential Time Hypothesis (ETH) fails - for any constant 0 <= sigma <= 1, there is p = Theta(k^sigma) such that there is no algorithm deciding in time 2^{o(sqrt{pk})} n^{O(1)} whether an n-vertex graph G can be transformed into a cluster graph with at most p cliques by changing at most k adjacencies. Thus, our upper and lower bounds provide an asymptotically tight analysis of the multivariate parameterized complexity of the problem for the whole range of values of p from constant to a linear function of k.

A Fast Branching Algorithm for Cluster Vertex Deletion

In the family of clustering problems we are given a set of objects (vertices of the graph), together with some observed pairwise similarities (edges). The goal is to identify clusters of similar objects by slightly modifying the graph to obtain a cluster graph (disjoint union of cliques). Hüffner et al. (Theory Comput. Syst. 47(1), 196–217, 2010) initiated the parameterized study of Cluster Vertex Deletion, where the allowed modification is vertex deletion, and presented an elegant 𝓞min(2kk6logk+n3,2kkmnlogn)

Breaking the 2n-barrier for Irredundance: Two lines of attack

\mathcal {O}\left (\min (2^{k} k^{6} \log k + n^{3}, 2^{k} km\sqrt {n} \log n)\right )