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For the vast majority of local graph problems standard dynamic programmingtechniques give c^tw V^O(1) algorithms, where tw is the treewidth of the inputgraph. On the other hand, for problems with a global requirement (usuallyconnectivity) the best-known algorithms were naive dynamic programming schemesrunning in tw^O(tw) V^O(1) time.  We breach this gap by introducing a technique we dubbed Cut&Count that allowsto produce c^tw V^O(1) Monte Carlo algorithms for most connectivity-typeproblems, including Hamiltonian Path, Feedback Vertex Set and ConnectedDominating Set, consequently answering the question raised by Lokshtanov, Marxand Saurabh [SODA'11] in a surprising way. We also show that (under reasonablecomplexity assumptions) the gap cannot be breached for some problems for whichCut&Count does not work, like CYCLE PACKING.  The constant c we obtain is in all cases small (at most 4 for undirectedproblems and at most 6 for directed ones), and in several cases we are able toshow that improving those constants would cause the Strong Exponential TimeHypothesis to fail.  Our results have numerous consequences in various fields, like FPTalgorithms, exact and approximate algorithms on planar and H-minor-free graphsand algorithms on graphs of bounded degree. In all these fields we are able toimprove the best-known results for some problems.

Solving connectivity problems parameterized by treewidth in single exponential time