Randomized Contractions Meet Lean Decompositions

The randomized contractions technique, introduced by Chitnis et al. in 2012, is a robust framework for designing parameterized algorithms for graph separation problems. On high level, an algorithm in this framework recurses on balanced separators while possible, and in the leaves of the recursion uses high connectivity of the graph at hand to highlight a solution by color coding. In 2014, a subset of the current authors showed that, given a graph $G$ and a budget $k$ for the cut size in the studied separation problem, one can compute a tree decomposition of $G$ with adhesions of size bounded in $k$ and with bags exhibiting the same high connectivity properties with respect to cuts of size at most $k$ as in the leaves of the recursion in the randomized contractions framework. This led to an FPT algorithm for the Minimum Bisection problem. In this paper, we provide a new construction algorithm for a tree decomposition with the aforementioned properties, by using the notion of lean decompositions of Thomas. Our algorithm is not only arguably simpler than the one from 2014, but also gives better parameter bounds; in particular, we provide best possible high connectivity properties with respect to edge cuts. This allows us to provide $2^{O(k \log k)} n^{O(1)}$-time parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner Multicut.