A Shifting Algorithm for Constrained min-max Partition on Trees

Abstract Let T = (V,E) be a rooted tree with n edges. We associate nonnegative weight w(v) and size s(v) with each vertex v in V. A q-partition of T into q connected components T1,T2,…,Tq is obtained by deleting k = q−1 edges of T, 1 ≤ k 1. Size-constrained min-max problem: Find a q-partition of T for which WP is a minimum over all partitions P satisfying S(Ti) ≤ M (M > 0). 2. Height-constrained min-max problem: Find a q-partition of T for which WP is a minimum over all partitions P satisfying height h(Ti) ≤ H (H is a positive integer). The first problem is shown to be NP-complete, while a polynomial algorithm is presented for the second problem.