Partitioning a Weighted Tree into Subtrees with Weights in a Given Range

Abstract

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are given integers such that 0 ≤ l ≤ u. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such a partition is called an (l, u)-partition. We deal with three problems to find an (l, u)-partition of a given graph: the minimum partition problem is to find an (l, u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l, u)-partition with a given number p of components. All these problems are NP-hard even for series-parallel graphs, but are solvable in linear time for paths. In this paper, we present the first polynomial-time algorithm to solve the three problems for arbitrary trees.

An Efficient Algorithm for Node-Weighted Tree Partitioning with Subtrees' Weights in a Given Range

“…The number of nodes for a generated tree is called the size of the tree. We compare the performance of our algorithm against the algorithm by Ito et al [3] in executing time. These algorithms are coded in C++ programming language and executed on a Pentium IV PC with 2.6 GHz CPU.…”

“…The proposed algorithm only needs to be modified slightly to solve the p- [l, u]partition problem. A series of experiments are conducted to evaluate our algorithm and compare it with the state-of-the-art polynomial-time algorithm presented by Ito et al [3]. Experimental results reveal that our proposed algorithm runs much faster than the algorithm presented by Ito et al and reduce the total computing time greatly in different cases.…”

“…In [8], a dynamic programming approach is proposed in minimizing the workload imbalance among metadata servers. Ito et al [3] present the [l, u]partition problem for trees and obtained a polynomial-time algorithm to solve p- [l, u]partition and minimum- [l, u]partition problems.…”

“…. , p} by means of dynamic programming similar to [1], [3], [5]. The details of computing S (T v , k) are described briefly as follows.…”

Partitioning a graph into connected components with fixed centers and optimizing cost‐based objective functions or equipartition criteria

“…Lucertini et al [29] presented a polynomial time algorithm for the most uniform problem on paths. Recently, Ito et al [20] found a polynomial time algorithm for the most uniform problem on trees. Perl and Schach [33] presented an elegant shifting algorithm for the max-min p-connected partition of a tree, and Becker and Perl [8] found a variant of this algorithm for the min-max problem.…”