Approximation algorithms for the maximally balanced connected graph tripartition problem

Given a vertex-weighted connected graph $$G = (V, E, w(\cdot ))$$, the maximally balanced connected graphk-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–mink-BGP (min–maxk-BGP, respectively), and has received much study since about four decades ago. On general graphs, max–mink-BGP is strongly NP-hard for every fixed $$k \ge 2$$, 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.