Approximation Algorithms for Maximally Balanced Connected Graph Partition

Given a connected graph G = (V, E) , we seek to partition the vertex set V into k non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these k parts is minimized. We refer this problem to as min-max balanced connected graph partition into k parts and denote it as k -BGP . The vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm. The vertex-weighted 2 -BGP and 3 -BGP admit a 5/4-approximation and a 3/2-approximation, respectively. When k ≥ 4 , no approximability result exists for k -BGP , i.e., the vertex unweighted variant, except a trivial k -approximation. In this paper, we present another 3/2-approximation for the 3 -BGP and then extend it to become a k /2-approximation for k -BGP , for any fixed k ≥ 3 . Furthermore, for 4 -BGP , we propose an improved 24/13-approximation. To these purposes, we have designed several local improvement operations, which could find more applications in related graph partition problems.