Jesper Nederlof

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.

Fast Polynomial-Space Algorithms Using Möbius Inversion: Improving on Steiner Tree and Related Problems

Given a graph with n vertices, k terminals and bounded integer weights on the edges, we compute the minimum Steiner Tree in

Saving space by algebraization

{\mathcal{O}^*}(2^k)

On Problems as Hard as CNF-SAT

time and polynomial space, where the

Inclusion/Exclusion Meets Measure and Conquer

{\mathcal{O}^*}

Fast Polynomial-Space Algorithms Using Inclusion-Exclusion

notation omits poly (n ,k ) factors. Among our results are also polynomial-space

Inclusion/Exclusion Branching for Partial Dominating Set and Set Splitting

\mathcal{O}^*(2^n)

Homomorphic hashing for sparse coefficient extraction

algorithms for several

Fast Zeta Transforms for Lattices with Few Irreducibles

{\mathcal{NP}}

Generalized graph clustering: recognizing (p, q)-cluster graphs

-complete spanning tree and partition problems. The previous fastest known algorithms for these problems use the technique of dynamic programming among subsets, and require exponential space. We introduce the concept of branching walks and extend the Inclusion-Exclusion algorithm of Karp for counting Hamiltonian paths. Moreover, we show that our algorithms can also be obtained by applying Mobius inversion on the recurrences used for the dynamic programming algorithms.