For the vast majority of local graph problems standard dynamic programming techniques give c tw |V | O(1) algorithms, where tw is the treewidth of the input graph. On the other hand, for problems with a global requirement (usually connectivity) the best-known algorithms were naive dynamic programming schemes running in tw O(tw) |V | O(1) time.We breach this gap by introducing a technique we dubbed Cut&Count that allows to produce c tw |V | O(1) Monte Carlo algorithms for most connectivity-type problems, including HAMILTONIAN PATH, FEEDBACK VERTEX SET and CONNECTED DOMINATING SET, consequently answering the question raised by Lokshtanov, Marx and Saurabh [SODA'11] in a surprising way. We also show that (under reasonable complexity assumptions) the gap cannot be breached for some problems for which Cut&Count does not work, like CYCLE PACKING.The constant c we obtain is in all cases small (at most 4 for undirected problems and at most 6 for directed ones), and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail.Our results have numerous consequences in various fields, like FPT algorithms, exact and approximate algorithms on planar and H-minor-free graphs and algorithms on graphs of bounded degree. In all these fields we are able to improve the best-known results for some problems.

We introduce a concept of parameterizing a problem above the optimum solution of its natural linear programming relaxation and prove that the node multiway cut problem is fixed-parameter tractable (FPT) in this setting. As a consequence we prove that node multiway cut is FPT, when parameterized above the maximum separating cut, resolving an open problem of Razgon.Our results imply O * (4 k ) algorithms for vertex cover above maximum matching and almost 2-SAT as well as an O * (2 k ) algorithm for node multiway cut with a standard parameterization by the solution size, improving previous bounds for these problems. ACM Reference Format:Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. 2013. On multiway cut parameterized above lower bounds.

In this paper we consider two above lower bound parameterizations of the Node Multiway 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 * (4 k ) algorithms for Vertex Cover above Maximum Matching and Almost 2-SAT as well as an O * (2 k ) algorithm for Node Multiway Cut with a standard parameterization by the solution size, improving previous bounds for these problems.

For the vast majority of local problems on graphs of small treewidth (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 treewidth 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 bridge 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 all these fields we are able to improve the best-known results for some problems. Also, looking from a more theoretical perspective, our results are surprising since the equivalence relation that partitions all partial solutions with respect to extendability to global solutions seems to consist of at least tw tw equivalence classes for all these problems. Our results answer an open problem raised by Lokshtanov, Marx and Saurabh [SODA’11]. In contrast to the problems aimed at minimizing 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 bridged for some problems that aim to maximize the number of connected components like Cycle Packing .

The classical FEEDBACK VERTEX SET problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. FEEDBACK VERTEX SET has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field.In this paper we consider a more general and difficult version of the problem, named SUBSET FEEDBACK VER-TEX SET (SUBSET-FVS in short) where an instance comes additionally with a set S ⊆ V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00].The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2 k n O(1) instances with the size of S bounded by O(k 3 ), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2 O(k log k) n O(1) time algorithm solving the SUBSET FEEDBACK VERTEX SET problem, proving that it is indeed fixed-parameter tractable.