The k-leaf spanning tree problem admits a klam value of 39

The klam value of an algorithm that runs in time O^∗(f(k)) is the maximal value k such that f(k)<10²⁰. Given a graph G and a parameter k, the k-LEAF SPANNING TREE (k-LST) problem asks if G contains a spanning tree with at least k leaves. This problem has been extensively studied over the past three decades. In 2000, Fellows et al. (2000) asked whether it admits a klam value of 50. A steady progress towards an affirmative answer continued until 5 years ago, when an algorithm of klam value 37 was discovered. Our contribution is twofold. First, we present an O^∗(3.188^k)-time parameterized algorithm for k-LST, which shows that the problem admits a klam value of 39. Second, we rely on an application of the bounded search trees technique where the correctness of rules crucially depends on the history of previously applied rules in a non-standard manner, encapsulated in a “dependency claim”. Similar claims may be used to capture the essence of other complex algorithms in a compact, useful manner.