The strength of a graph is a measure of its vulnerability, strictly generalizing the edge connectivity of a weighted graph. Cunningham [J. Assoc. Comput. Mach., 32 (1985), pp. 549–562] showed how to compute the strength of a graph in $O(| V |^4 | E |)$ time, using ideas from polymatroids and network flow. In this paper, his polymatroid approach is modified, a modified version of the Goldberg-Tarjan network flow algorithm [J Assoc. Comput. Mach., 4 (1988), pp. 136-146] is used. Then, using ideas developed by Gallo, Grigoriadis, and Tarjan [SIAM J. Comput., 18 (1989), pp. 30–55], and by Gusfield, Martel, and Fernandez-Baca in [SIAM J. Comput.,16 (1987), pp. 237–251], it is shown that the solution runs in $O(| V |^3 | E |)$ time. Analogous speedups for sparse-case bounds are also obtainable.
[1]
Robert W. Irving,et al.
The Stable marriage problem - structure and algorithms
,
1989,
Foundations of computing series.
[2]
William H. Cunningham,et al.
Optimal attack and reinforcement of a network
,
1985,
JACM.
[3]
Andrew Vladislav Goldberg,et al.
Efficient graph algorithms for sequential and parallel computers
,
1987
.
[4]
Andrew V. Goldberg,et al.
A new approach to the maximum flow problem
,
1986,
STOC '86.
[5]
To-Yat Cheung.
Multifacility Location Problem with Rectilinear Distance by the Minimum-Cut Approach
,
1980,
TOMS.
[6]
Charles U. Martel,et al.
Fast Algorithms for Bipartite Network Flow
,
1987,
SIAM J. Comput..