Spanning Trees with Many Leaves in Regular Bipartite Graphs

Given a d-regular bipartite graph Gd, whose nodes are divided in black nodes and white nodes according to the partition, we consider the problem of computing the spanning tree of Gd with the maximum number of black leaves. We prove that the problem is NP hard for any fixed d ≥ 4 and we present a simple greedy algorithm that gives a constant approximation ratio for the problem. More precisely our algorithm can be used to get in linear time an approximation ratio of 2 - 2/(d - 1)2 for d ≥ 4. When applied to cubic bipartite graphs the algorithm only achieves a 2-approximation ratio. Hence we introduce a local optimization step that allows us to improve the approximation ratio for cubic bipartite graphs to 1.5. Focusing on structural properties, the analysis of our algorithm proves a lower bound on lB(n, d), i.e., the minimum m such that every Gd with n black nodes has a spanning tree with at least m black leaves. In particular, for d = 3 we prove that lB(n, 3) is exactly ⌈n/3⌉ +1.