On Out-Trees With Many Leaves

The k-LEAF OUT-BRANCHING problem is to find an out-branching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the k-LEAF-OUTBRANCHING problem. We give the first polynomial kernel for ROOTED k-LEAF-OUT-BRANCHING, a variant of k-LEAF-OUT-BRANCHING where the root of the tree searched for is also a part of the input. Our kernel with O(k3) vertices is obtained using extremal combinatorics. For the k-LEAF-OUT-BRANCHING problem, we show that no polynomial-sized kernel is possible unless coNP is in NP/poly. However, our positive results for ROOTED k-LEAF-OUT-BRANCHING immediately imply that the seemingly intractable k-LEAF-OUT-BRANCHING problem admits a data reduction to n independent polynomial-sized kernels. These two results, tractability and intractability side by side, are the first ones separating Karp kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.