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 {\em
min-max balanced connected graph partition} into k parts and denote it as
{\sc 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 {\sc 2-BGP} and {\sc 3-BGP} admit a
5/4-approximation and a 3/2-approximation, respectively; but no
approximability result exists for {\sc k-BGP} when k≥4, except a
trivial k-approximation. In this paper, we present another
3/2-approximation for our cardinality {\sc 3-BGP} and then extend it to
become a k/2-approximation for {\sc k-BGP}, for any constant k≥3.
Furthermore, for {\sc 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.Comment: 23 pages, 7 figures, accepted for presentation at COCOA 2019 (Xiamen,
China