Abstract
In this paper, we consider the k-sink location problem in a path network with the goal of optimizing a combinational function of the maximum completion time and the total completion time. Let \(P=\left( V,E\right) \) be an undirected path network with n vertices. Each vertex has a positive weight, indicating the initial amount of supplies, and each edge has a positive length and a uniform capacity, which is the maximum amount of supplies that can enter the edge per unit time. Our goal is to identify k sink locations on the path P so that all supplies will be successfully evacuated and the given objective function is optimized. This paper presents two efficient polynomial time algorithms, which achieve \(O\left( n \right) \) for \(k=1\) and \(O\left( n^6 \right) \) for general k, respectively.
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Acknowledgements
This research was supported by the MOE (Ministry of Education in China) Project of Humanities and Social Sciences under Grant 18YJC630114, the National Natural Science Foundation of China under Grant 71701162, the Fundamental Research Funds for the Central Universities under Grant XJS190602, the Natural Science Foundation of Shaanxi Province of China under Grants 2019JQ-079, 2019JQ-154, 2015JQ7278, and the China Postdoctoral Science Foundation under Grant 2017M613192.
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Appendices
Proof of Lemma 2
Proof
Compute \(f_1^L\left( v_i\right) \) in increasing order of i. By definition, \(f_1^L\left( v_1\right) =0\). According to Eq. (2), \(f_1^L\left( v_2\right) =\left( v_2-v_1\right) \tau + w_1/c\). For \(2 < i \le n\), we have
Therefore, \(f_1^L\left( v_i\right) \) for all \(v_i\) can be computed in \(O\left( n\right) \) time. Because of the symmetry of \(f_1^L\left( v_i\right) \) and \(f_1^R\left( v_i\right) \), \(f_1^R\left( v_i\right) \) for all \(v_i\) can also be computed similarly in reverse order of i. The calculation is as follows:
\(\square \)
Proof of Lemma 3
Proof
We focus on the calculation of \(f_2^L \left( v_i\right) \), without loss of generality, because \(f_2^R \left( v_i\right) \) can be computed in a similar manner. Compute \(f_2^L\left( v_i\right) \) in increasing order of i. By definition, \(f_2^L\left( v_1\right) =0\). According to Eq. (4), \(f_2^L\left( v_2\right) =\left( v_2-v_1\right) w_1 \tau + {w_1}^2/\left( 2c\right) \). Below, we show how to compute \(f_2^L\left( v_{i+1}\right) \) in constant time, given \(f_2^L\left( v_{j}\right) \), \(j \le i\).
Different sink points usually mean different congestion situations, that is, different \(P^\prime \). For convenience, let \(P_i^\prime \) be the new path, where \(v_i\) is the sink point. Therefore, we need to obtain n new paths. We show how to transform P to \(P^\prime \) in \(O\left( n\right) \) time. The transformation of the left side of the sink point \(v_i\) is taken as an example, and the right side can be treated in the same manner. In the following, \(P_i^\prime \) is always used to mean the left side of \(P_i^\prime \). To determine whether a vertex \(v_{i-1}\) will congest at vertex \(v_{i}\), we actually need to know whether \(\left( v_{i}-v_{i-1} \right) \tau \) is greater than \(w_{i}/c\). \(\left( v_{i}-v_{i-1} \right) \tau \ge w_{i}/c\) means that \(v_{i-1}\) does not need to wait at \(v_{i}\). In this case, the two vertices can keep their respective weights. \(\left( v_{i}-v_{i-1} \right) \tau < w_{i}/c\) means that \(v_{i-1}\) should wait at \(v_{i}\), that is, congestion occurs. In this case, \(v_{i-1}\) and \(v_{i}\) are merged into a single new vertex, whose weight is \(\left( w_i+w_{i-1}\right) \). If the sink point is \(v_i\), \(v_{i-1}\) will never congest at \(v_i\). In \(O\left( n\right) \) time, we can determine whether \(v_{i-1}\) will congest at \(v_{i}\), for all \(2\le i \le n\).
Without loss of generality, suppose that
If \(v_{i-1}\) will not congest at \(v_i\), we have the following recursive formula:
Otherwise, if \(v_{i-1}\) will congest at \(v_{i}\), then \(v_{i-1}\) and \(v_i\) should be merged as a new vertex. In this case, we also have the following recursive formula,
Therefore, we can obtain all \(f_2^L\left( v_i\right) \) within \(O\left( n\right) \) time. The lemma follows. \(\square \)
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Luo, T., Li, H., Ru, S. et al. Multiple sink location problem in path networks with a combinational objective. Optim Lett 15, 733–755 (2021). https://doi.org/10.1007/s11590-020-01597-w
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DOI: https://doi.org/10.1007/s11590-020-01597-w