A Response to Volgenant’s Addendum
on the Most
Chun-Nan Hung and Lih-Hsing Hsu*
Department of Computer and Information Science, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
Institute of Information Science, Academia Sinica, Taipei, Taiwan, Republic of China
Let G = ( U U V, E ) be a weighted bipartite graph having an edge weight u’, 2 0 for each e in E . An edge is
called a must vitul edge if its removal from G results in
the largest decrease in the total weight of the maximum weighted matching. In [ I ] , an O( n 3 ) algorithm was pre- sented to obtain the most vital edges. In [ 3 ] , Volgenant pointed out that the most vital edges can also be found using the driul solutions of the linear assignment problem [ 21. We were unaware of this result and studied this prob- lem from different point of view. In our paper, we first gave characterization of the most vital edges in Lemma 1 and the effect of deleting a matched edge which is a candidate for the most vital edges in Lemma 6. Our mo- tivation was to study the effect on the cost of any com- binatonal optimization problem subject to the deletion of an edge in turn.
* To whom correspondence should be addressed
NETWORKS, Vol. 27 (1 996) 255
(c 1996 John Wiley & Sons, Inc.
We did not necessarily know the shortest distances ui
from an arbitrarily chosen vertex to all other vertices i,
i.e., the dual solution specified in [ 2 ] . Thus, we simply chose Floyd’s algorithm rather than Dijkstra’s algorithm to accommodate negative edge weights for solving short- est-path problems.
[ 1 1 C. N. Hung, L. H. Hsu, and T. Y. Sung, The most vital edges of matchings in a bipartite graph, Networks 23 ( 1993) 309-3 13.
G. Kindervater, A. Volgenant, G. de Leve, and V. van Gijlswijk, On dual solutions of the linear assignment problem. Eur. J. of Oper. Rex 19 (1985) 76-81.
A. Volgenant, An addendum to the most vital edges of matching in a bipartite graph. Networks, to appear. [ 2 ]
[ 3 ]
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