Abstract
This paper presents a strongly polynomial time algorithm for the minimum cost tension problem, which runs in \(O(\max \{m^3n, m^2 \log n(m+n \log n)\})\) time, where n and m denote the number of nodes and number of arcs, respectively. Our algorithm improves upon the previous strongly polynomial time of \(O(n^4 m^3 \log n)\) due to Hadjiat and Maurras (Discret Math 165(166):377–394, 1997).
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Appendix
Appendix
The Proof of Lemma 3.9
By Definition 3.4 and the assumption of this lemma, we have \(\Delta x_{ij}>0\), so, by \(x^{\prime }_{ij}=x_{ij}+\alpha \Delta x_{ij}\) and \(\alpha \le \alpha _2\), we get
We have \(\Delta x_{ij}>0\) and \(c_{ij}-x_{ij}\ge 0\), which means \(\frac{\Delta x_{ij}}{1+ \Delta x_{ij}}<1\) and \(\frac{\Delta x_{ij}}{1+ \Delta x_{ij}}(c_{ij}-x_{ij})\le c_{ij}-x_{ij}\), or
By Lemma 3.6, \(1+\Delta x_{ij}\le 2m-2< 2m\). Thus, by \(\Delta x_{ij}>0\), we get \(\frac{1}{1+\Delta x_{ij}}\ge \frac{1}{2m}\). Hence, by (9) and (10),
On the other hand, by definitions, we have \(c_{ij}-x_{ij}\le \delta _{max}(\theta ^{\prime }, x)\), or \(c_{ij} - \delta _{max}(\theta ^{\prime }, x) \le x_{ij}\). All data are integer, so \(\Delta x_{ij}\ge 1\), which means, by Lemma 3.7,
Hence,
Therefore, by (11), \( |\delta (\theta ^{\prime },x^{\prime })_{ij}| \le (1-\frac{1}{2m})\delta _{max}(\theta ^{\prime }, x)\). \(\square \)
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Ghiyasvand, M. A faster strongly polynomial time algorithm to solve the minimum cost tension problem. J Comb Optim 34, 203–217 (2017). https://doi.org/10.1007/s10878-016-0070-4
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DOI: https://doi.org/10.1007/s10878-016-0070-4