Designing FPT Algorithms for Cut Problems Using Randomized Contractions

We introduce a new technique for designing fixed-parameter algorithms for cut problems, called randomized contractions. We apply our framework to obtain the first fixed-parameter algorithms (FPT algorithms) with exponential speed up for the Steiner Cut and Node Multiway Cut-Uncut problems. 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 $2^{O(k^2\log |\Sigma|)}n^4\log n$ deterministic time (even in the stronger, vertex-deletion variant), where $k$ is the number of unsatisfied edges and $|\Sigma|$ 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 $O(\sqrt{\log n})$ 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 and $\tilde{O}(2^{O(k^2\log k)}n^2)$ randomized 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 presented at 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 [ACM Trans. Algorithms, 9 (2013), 30], which yields algorithms with double exponential running time. An interesting aspect of our algorithms is that they can handle positive real weights.