3 research outputs found
Approximation algorithms for maximally balanced connected graph partition
Given a simple connected graph , we seek to partition the vertex
set into 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 parts is minimized. We refer this problem to as {\em
min-max balanced connected graph partition} into parts and denote it as
{\sc -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 -BGP} and {\sc -BGP} admit a
-approximation and a -approximation, respectively; but no
approximability result exists for {\sc -BGP} when , except a
trivial -approximation. In this paper, we present another
-approximation for our cardinality {\sc -BGP} and then extend it to
become a -approximation for {\sc -BGP}, for any constant .
Furthermore, for {\sc -BGP}, we propose an improved -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
Approximation Algorithms for Minimum Tree Partition
We consider a problem of locating communication centers. In this problem, it is required to partition the set of n customers into subsets minimizing the length of nets required to connect all the customers to the communication centers. Suppose that communication centers are to be placed in p of the customers locations. The number of customers each center supports is also given. The problem remains to divide a graph into sets of the given sizes, keeping the sum of the spanning trees minimal. The problem is NP-Complete, and no polynomial algorithm with bounded error ratio can be given, unless P = NP . We present an approximation algorithm for the problem assuming that the edge lengths satisfy the triangle inequality. It runs in O(p 2 4 p +n 2 ) time (n = jV j) and comes within a factor of 2p \Gamma 1 of optimal. When the sets' sizes are all equal this algorithm runs in O(n 2 ) time. Next an improved algorithm is presented which obtains as an input a positive integer x (x n \Gamm..