Approximation algorithms for the maximally balanced connected graph tripartition problem

Given a vertex-weighted connected graph G = ( V , E , w ( · ) ) , the maximally balanced connected graph k - partition ( k - BGP ) seeks to partition the vertex set V into k non-empty parts such that the subgraph induced by each part is connected and the weights of these k parts are as balanced as possible. When the concrete objective is to maximize the minimum (to minimize the maximum, respectively) weight of the k parts, the problem is denoted as max – min k - BGP ( min – max k - BGP , respectively), and has received much study since about four decades ago. On general graphs, max – min k - BGP is strongly NP-hard for every fixed k ≥ 2 , and remains NP-hard even for the vertex uniformly weighted case; when k is part of the input, the problem is denoted as max – min BGP , and cannot be approximated within 6/5 unless P = NP. In this paper, we study the tripartition problems from approximation algorithms perspective and present a 3/2-approximation for min – max 3- BGP and a 5/3-approximation for max – min 3- BGP , respectively. These are the first non-trivial approximation algorithms for 3- BGP , to our best knowledge.