K-distinct branchings admits a polynomial kernel

Unlike the problem of deciding whether a digraph D = (V, A) has ℓ in-branchings (or ℓ out-branchings) is polynomial time solvable, the problem of deciding whether a digraph D = (V, A) has an in-branching B⁻ and an out-branching B⁺ which are arc-disjoint is NP-complete. Motivated by this, a natural optimization question that has been studied in the realm of Parameterized Complexity is called Rooted k-Distinct Branchings. In this problem, a digraph D = (V, A) with two prescribed vertices s, t are given as input and the question is whether D has an in-branching rooted at t and an out-branching rooted at s such that they differ on at least k arcs. Bang-Jensen et al. [Algorithmica, 2016 ] showed that the problem is fixed parameter tractable (FPT) on strongly connected digraphs. Gutin et al. [ICALP, 2017; JCSS, 2018 ] completely resolved this problem by designing an algorithm with running time 2^O(^{k2 log2 k)}n^O(1). Here, n denotes the number of vertices of the input digraph. In this paper, answering an open question of Gutin et al., we design a polynomial kernel for Rooted k-Distinct Branchings. In particular, we obtain the following: Given an instance (D, k, s, t) of Rooted k-Distinct Branchings, in polynomial time we obtain an equivalent instance (D^′, k^′, s, t) of Rooted k-Distinct Branchings such that |V (D^′)| ≤ O(k²) and the treewidth of the underlying undirected graph is at most O(k). This result immediately yields an FPT algorithm with running time 2^O(k log k⁾ + n^O(1); improving upon the previous running time of Gutin et al. For our algorithms, we prove a structural result about paths avoiding many arcs in a given in-branching or out-branching. This result might turn out to be useful for getting other results for problems concerning in-and out-branchings.