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A sifting-edges algorithm for accelerating the computation of absolute 1-center in graphs

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Abstract

Let \(G = (V, E, w)\) be an undirected connected edge-weighted graph, where V is the n-vertices set, E is the m-edges set, and \(w: E \rightarrow \mathbb {R}^+\) is a positive edge-weight function. Given \(G = (V, E, w)\), a subset \(X \subseteq V\) of p terminals, and a subset \(F \subseteq E\) of candidate edges, the Absolute 1-Center Problem (A1CP) aims to find a point on some edge in F to minimize the distance from it to X. This paper revisits this classic and polynomial-time solvable problem from a novel perspective and finds some new and nontrivial properties of it, with the highlight of establishing the relationship between the A1CP and the saddle point of distance matrix. In this paper, we prove that an absolute 1-center is just a vertex 1-center if the all-pairs shortest paths distance matrix from the vertices covered by the candidate edges in F to X has a (global) saddle point. Furthermore, we define the local saddle point of edge and demonstrate that we can sift the candidate edge having a local saddle point. By incorporating the method of sifting edges into the framework of the well-known Kariv and Hakimi’s algorithm, we develop an \(O(m + p m^*+ n p \log p)\)-time algorithm for A1CP, where \(m^*\) is the number of the remaining candidate edges. Specifically, it takes \(O(m^*n + n^2 \log n)\) time to apply our algorithm to the classic A1CP when the distance matrix is known and \(O(m n + n^2 \log n)\) time when the distance matrix is unknown, which are smaller than \(O(mn + n^2 \log n)\) and \(O(mn + n^3)\) of Kariv and Hakimi’s algorithm, respectively.

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References

  • Aboutahoun AW, EI-Safty F (2020) Optimal algorithms for weighted \(1\)-center problem in deterministic and stochastic tree networks. Alex Eng J 59:3463–3471

    Article  Google Scholar 

  • Ding W, Qiu K (2014) Algorithms for the minimum diameter terminal steiner tree problem. J Comb Optim 28(4):837–853

    Article  MathSciNet  Google Scholar 

  • Ding W, Qiu K (2017) An FPTAS for generalized absolute \(1\)-center problem in vertex-weighted graphs. J Comb Optim 34:1084–1095

    Article  MathSciNet  Google Scholar 

  • Ding W, Qiu K (2019) Sifting edges to accelerate the computation of absolute \(1\)-center in graphs. In: Proceedings of WCGO 2019, pp 468–476

  • Eiselt HA, Marianov V (2011) Foundations of location analysis. Springer, Heidelberg

    Book  Google Scholar 

  • Fredman ML, Tarjan RE (1987) Fibonacci heaps and their uses in improved network optimization algorithms. J ACM 34(3):596–615

    Article  MathSciNet  Google Scholar 

  • Goldman AJ (1972) Minimax location of a facility in a network. Transp Sci 6(4):407–418

    Article  MathSciNet  Google Scholar 

  • Hakimi SL (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Oper Res 12(3):450–459

    Article  Google Scholar 

  • Hakimi SL, Schmeichel EF, Pierce JG (1978) On \(p\)-centers in networks. Transp Sci 12(1):1–15

    Article  MathSciNet  Google Scholar 

  • Handler GY (1973) Minimax location of a facility in an undirected tree graph. Transp. Sci. 7(3):287–293

    Article  MathSciNet  Google Scholar 

  • Hassin R, Tamir A (1995) On the minimum diameter spanning tree problem. Info Proc Lett 53(2):109–111

    Article  MathSciNet  Google Scholar 

  • Karger DR, Koller D, Phillips SJ (1993) Finding the hidden path: time bounds for all-pairs shortest paths. SIAM J Comput 22(6):1199–1217

    Article  MathSciNet  Google Scholar 

  • Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems. I: the \(p\)-centers. SIAM J Appl Math 37(3):513–538

    Article  MathSciNet  Google Scholar 

  • Pettie S (2004) A new approach to all-pairs shortest paths on real-weighted graphs. Theor Comp Sci 312(1):47–74

    Article  MathSciNet  Google Scholar 

  • Santiváñez J, Melachrinoudis E (2007) Location of a reliable center on a tree network. Oper Res 7(3):419–445

    Google Scholar 

  • Santiváñez J, Melachrinoudis E, Helander ME (2009) Network location of a reliable center using the most reliable route policy. Comput Oper Res 36(5):1437–1460

    Article  Google Scholar 

  • Tansel BC, Francis RL, Lowe TJ (1983) Location on networks: a survey. Part I: the \(p\)-center and \(p\)-median problems. Manag Sci 29(4):482–497

    Article  Google Scholar 

  • Xue G (1997) Linear time algorithms for computing the most reliable source on an unreliable tree network. Networks 30:37–45

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author and the fourth author were supported by the National Natural Science Foundation of China (No. 51979249). The third author was supported by the National Natural Science Foundation of China (No. 62002255) and the Science and Technology Innovation Project for Colleges and Universities in Shanxi Province (No. 2019L0353).

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Correspondence to Wei Ding.

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An extended abstract of this paper appeared in the Proceedings of the 6th World Congress on Global Optimization (WCGO’19) (Ding and Qiu 2019)

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Ding, W., Qiu, K., Zhou, Y. et al. A sifting-edges algorithm for accelerating the computation of absolute 1-center in graphs. J Comb Optim 44, 905–920 (2022). https://doi.org/10.1007/s10878-022-00866-x

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