On maximum leaf trees and connections to connected maximum cut problems

In an instance of the (directed) Max Leaf Tree (MLT) problem we are given a vertexweighted (di)graph G(V,E,w) and the goal is to compute a subtree with maximum weight on the leaves. The weighted Connected Max Cut (CMC) problem takes in an undirected edgeweighted graph G(V,E,w) and seeks a subset S ⊆ V such that the induced graph G[S] is connected and ∑ e∈δ(S)w(e) is maximized. We obtain a constant approximation algorithm for MLT when the weights are chosen from {0, 1}, which in turn implies a Ω(1/ log n) approximation for the general case. We show that the MLT and CMC problems are related and use the algorithm for MLT to improve the factor for CMC from Ω(1/ log n) (Hajiaghayi et al., ESA 2015) to Ω(1/ log n).