Designing FPT algorithms for cut problems using randomized contractions

We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely 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 2O(k 2 log |Σ|)n4 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