Marek Cygan

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.

Exponential-time approximation of weighted set cover

The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponential-time algorithms for problems of that kind. The goal is getting the time complexity still of order O(c^n), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation. In fact, we describe a kind of approximation schemes-trade-offs between approximation factor and the time complexity. We use general transformations from exponential-time exact algorithms to approximations that are faster but still exponential-time. For example, we show that for any reduction rate r, one can transform any O^*(c^n)-time^1 algorithm for Set Cover into a (1+lnr)-approximation algorithm running in time O^*(c^n^/^r). We believe that results of that kind extend the applicability of exact algorithms for NP-hard problems.

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.

On Problems as Hard as CNF-SAT

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2^n), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2^n), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon

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.

Algorithmic Applications of Baur-Strassen’s Theorem: Shortest Cycles, Diameter, and Matchings

Consider a directed or undirected graph with integral edge weights in [-W, W]. This paper introduces a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the Baur-Strassen Theorem and Strojohanns determinant algorithm. For directed and undirected graphs without negative cycles we obtain simple Õ(Wnω) running time algorithms for finding a shortest cycle, computing the diameter or radius, and detecting a negative weight cycle. For each of these problems we unify and extend the class of graphs for which Õ(Wnω) time algorithms are known. In particular no such algorithms were known for any of these problems in undirected graphs with (potentially) negative weights. We also present an Õ(Wnω) time algorithm for minimum weight perfect matching. This resolves an open problem posed by Sankowski in 2006, who presented such an algorithm for bipartite graphs. Our algorithm uses a novel combinatorial interpretation of the linear program dual for minimum perfect matching. We believe this framework will find applications for finding larger spectra of related problems. As an example we give a simple Õ(Wnω) time algorithm to find all the vertices that lie on cycles of length at most t, for given t. This improves an Õ(Wnω) time algorithm of Yuster.

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.

Parameterized complexity of firefighting revisited

The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and the fire spreads to its neighborhood in the graph in so far no firefighters are protecting those vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to bipartite graphs or to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. We complete these results by showing that the problem is in FPT on general graphs when parameterized by the number of burned vertices, but has no polynomial kernel on trees, resolving an open problem. Conversely, we show that the problem is W[1]-hard when parameterized by the number of unburned vertices, even on bipartite graphs. For both parameterizations, we additionally give refined algorithms on trees, improving on the running times of the known algorithms.

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)