Approximation Algorithms for Maximally Balanced Connected Graph Partition.

Given a simple 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 general 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; but no approximability result exists for k -BGP when \(k \ge 4\), except a trivial k-approximation. In this paper, we present another 3/2-approximation for our cardinality 3-BGP and then extend it to become a k/2-approximation for k -BGP, for any constant \(k \ge 3\). Furthermore, for 4-BGP, we propose an improved 24/13-approximation. To these purposes, we have designed several local improvement operations, which could be useful for related graph partition problems.