No abstract

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 new technique for designing fixed-parameter algorithms for cut problems, called randomized contractions. We apply our framework to obtain the first FPT algorithm for the UNIQUE LABEL COVER problem and new FPT algorithms with exponential speed up for the STEINER CUT and NODE MULTIWAY CUT-UNCUT problems. More precisely, we show the following:• We prove that the parameterized version of the UNIQUE LABEL COVER problem, which is the base of the UNIQUE GAMES CONJECTURE, can be solved in 2 O(k 2 log |Σ|) n 4 log n deterministic time (even in the stronger, vertex-deletion variant) where k is the number of unsatisfied edges and |Σ| is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of UNIQUE GAMES where the number of edges allowed not to be satisfied is upper bounded by O( √ log n) to optimality, which improves over the trivial O(1) upper bound.• We prove that the STEINER CUT problem can be solved in 2 O(k 2 log k) n 4 log n deterministic time andrandomized time where k is the size of the cutset. This result improves the double exponential running time of the recent work of Kawarabayashi and Thorup (FOCS'11).• We show how to combine considering 'cut' and 'uncut' constraints at the same time. More precisely, we define a robust problem NODE MULTIWAY CUT-UNCUT that can serve as an abstraction of introducing uncut constraints, and show that it admits an algorithm running in 2 O(k 2 log k) n 4 log n deterministic time where k is the size of the cutset. To the best of our knowledge, the only known way of tackling uncut constraints was via the approach of Marx, O'Sullivan and Razgon (STACS'10, ACM Trans. Alg. 2013), which yields algorithms with double exponential running time.An interesting aspect of our algorithms is that they can handle positive real weights.

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.

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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 .

We present two new deterministic algorithms for the Feedback Vertex Set problem parameterized by the solution size. We begin with a simple algorithm, which runs in O * ((2 + φ) k ) time, where φ < 1.619 is the golden ratio. It already surpasses the previously fastest O * ((1 + 2 √ 2) k )-time deterministic algorithm due to Cao et al. [SWAT 2010]. In our developments we follow the approach of Cao et al., however, thanks to a new reduction rule, we obtain not only better dependency on the parameter in the running time, but also a solution with simple analysis and only a single branching rule. Then, we present a modification of the algorithm which, using a more involved set of branching rules, achieves O * (3.592 k ) running time.