A Network Improvement Problem Under Different Norms

Authors:
J. Z. Zhang;X. G. Yang;M. C. Cai
Affiliations:
Department of Mathematics, City University of Hong Kong, Hong Kong;Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China;Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China
Venue:
Computational Optimization and Applications
Year:
2004

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Abstract

In this paper, we first consider a network improvement problem, called vertex-to-vertices distance reduction problem. The problem is how to use a minimum cost to reduce lengths of the edges in a network so that the distances from a given vertex to all other vertices are within a given upper bound. We use l∞, l1 and l2 norms to measure the total modification cost respectively. Under l∞ norm, we present a strongly polynomial algorithm to solve the problem, and under l1 or weighted l2 norm, we show that achieving an approximation ratio O(log(|V|)) is NP-hard. We also extend the results to the vertex-to-points distance reduction problem, which is to reduce the lengths of edges most economically so that the distances from a given vertex to all points on the edges of the network are within a given upper bound.