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.

Authors

• Marek Cygan
• Jesper Nederlof
• Marcin Pilipczuk
• Michal Pilipczuk
• Johan M. M. van Rooij
• Jakub Onufry Wojtaszczyk

Keywords

• Approximation algorithms
• Heuristic algorithms
• Dynamic programming
• Educational institutions
• Steiner trees
• Monte Carlo methods
• Polynomials
• Exponential Time
• Single Exponential Time
• Estimation Algorithm
• Standard Program
• Algorithm For Problem
• Monte Carlo Algorithm
• Exact Algorithm
• Strong Hypothesis
• Dominating Set
• Hamiltonian Path
• Left Side
• Lower Bound
• Running Time
• Variety Of Settings
• Undirected
• Parametrized
• Weight Function
• Even Number
• Vertices
• Vertex Cover
• Counting Technique
• Compression Step
• Space Of Polynomials
• Candidate Solutions
• Traveling Salesman Problem
• Joining Tree
• Algorithm Running
• Graph Algorithms
• treewidth
• ? xed parameter tractability
• randomized algorithms
• exact algorithms