Sum-of-Max Chain Partition of a Tree

Path partition problems on trees have found various applications. In this paper, we present an $O(n \log n)$ time algorithm for solving the following variant of path partition problem: given a rooted tree of $n$ nodes $1, \ldots, n$, where vertex $i$ is associated with a weight $w_i$ and a cost $s_i$, partition the tree into several disjoint chains $C_1,\ldots,C_k$, so that the weight of each chain is no more than a threshold $w_0$ and the sum of the largest $s_i$ in each chain is minimized. We also generalize the algorithm to the case where the cost of a chain is determined by the $s_i$ of the vertex with the highest rank in the chain, which can be determined by an arbitrary total order defined on all nodes instead of the value of $s_i$.