FPTASs for trimming weighted trees

Abstract

Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number $k\ge 0$, we consider the following four cut problems: cutting vertices of weight at most or at least $k$ from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The MinMstCut problem (cut vertices of weight at most $k$ and minimize the total cost of the edges being deleted) can be solved in linear time and space and the other three problems are NP-hard. In this paper, we design an $O(nl/\epsilon )$-time $O(l^2/\epsilon +n)$-space algorithm for MaxMstCut, and $O\left(nl(1/\epsilon +logn)\right)$-time $O(l^2/\epsilon +n)$-space algorithms for the other two problems, MinLstCut and MaxLstCut, where $n$ is the number of vertices in the tree, $l$ the number of leaves, and $\epsilon >0$ the prescribed error bound.