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 ∞, l 1 and l 2 norms to measure the total modification cost respectively. Under l ∞ norm, we present a strongly polynomial algorithm to solve the problem, and under l 1 or weighted l 2 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.
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Zhang, J., Yang, X. & Cai, M. A Network Improvement Problem Under Different Norms. Computational Optimization and Applications 27, 305–319 (2004). https://doi.org/10.1023/B:COAP.0000013061.17529.79
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DOI: https://doi.org/10.1023/B:COAP.0000013061.17529.79