TY - JOUR

T1 - Approximation algorithms for minimum tree partition

AU - Guttmann-Beck, Nili

AU - Hassin, Refael

PY - 1998/10/5

Y1 - 1998/10/5

N2 - 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(p24p + n2) time (n = |V|) and comes within a factor of 2p - 1 of optimal. When the sets' sizes are all equal this algorithm runs in O(n2) time. Next, an improved algorithm is presented which obtains as an input a positive integer x (x≤n - p) and runs in O(f(p,x)n2) time, where f is an exponential function of p and x, and comes within a factor of 2 + (2p - 3)/x of optimal. When the sets' sizes are all equal it runs in O(2(p+x)n2) time. A special algorithm is presented for the case p = 2.

AB - 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(p24p + n2) time (n = |V|) and comes within a factor of 2p - 1 of optimal. When the sets' sizes are all equal this algorithm runs in O(n2) time. Next, an improved algorithm is presented which obtains as an input a positive integer x (x≤n - p) and runs in O(f(p,x)n2) time, where f is an exponential function of p and x, and comes within a factor of 2 + (2p - 3)/x of optimal. When the sets' sizes are all equal it runs in O(2(p+x)n2) time. A special algorithm is presented for the case p = 2.

KW - Approximation algorithms

KW - Graph partitioning

KW - Minimum spanning tree

UR - http://www.scopus.com/inward/record.url?scp=0042722346&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(98)00052-3

DO - 10.1016/S0166-218X(98)00052-3

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AN - SCOPUS:0042722346

SN - 0166-218X

VL - 87

SP - 117

EP - 137

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 1-3

ER -