Approximation Algorithms for Maximally Balanced Connected Graph Partition

• Published in: Algorithmica Vol. 83; no. 12; pp. 3715 - 3740
• Main Authors: Chen, Yong, Chen, Zhi-Zhong, Lin, Guohui, Xu, Yao, Zhang, An
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2021; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Approximation; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph theory; Mathematical analysis; Mathematics of Computing; Partitions (mathematics); Theory of Computation; Vertex sets; Graph partition; Approximation algorithm; Induced subgraph; Connected component; Local improvement
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-021-00870-3

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.