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Inverse max+sum spanning tree problem under weighted $$l_{\infty }$$ l ∞ norm by modifying max-weight vector

Author

Listed:
  • Junhua Jia

    (Southeast University)

  • Xiucui Guan

    (Southeast University)

  • Qiao Zhang

    (Southeast University)

  • Xinqiang Qian

    (Southeast University)

  • Panos M. Pardalos

    (University of Florida
    Higher School of Economics)

Abstract

The max+sum spanning tree (MSST) problem is to determine a spanning tree T whose combined weight $$\max _{e\in T}w(e)+\sum _{e\in T}c(e)$$ max e ∈ T w ( e ) + ∑ e ∈ T c ( e ) is minimum for a given edge-weighted undirected network G(V, E, c, w). This problem can be solved within $$O(m \log n)$$ O ( m log n ) time, where m and n are the numbers of edges and nodes, respectively. An inverse MSST problem (IMSST) aims to determine a new weight vector $$\bar{w}$$ w ¯ so that a given $$T^0$$ T 0 becomes an optimal MSST of the new network $$G(V,E,c,\bar{w})$$ G ( V , E , c , w ¯ ) . The IMSST problem under weighted $$l_\infty $$ l ∞ norm is to minimize the modification cost $$\max _{e\in E} q(e)|\bar{w}(e)-w(e)|$$ max e ∈ E q ( e ) | w ¯ ( e ) - w ( e ) | , where q(e) is a cost modifying the weight w(e). We first provide some optimality conditions of the problem. Then we propose a strongly polynomial time algorithm in $$O(m^2\log n)$$ O ( m 2 log n ) time based on a binary search and a greedy method. There are O(m) iterations and we solve an MSST problem under a new weight in each iteration. Finally, we perform some numerical experiments to show the efficiency of the algorithm.

Suggested Citation

  • Junhua Jia & Xiucui Guan & Qiao Zhang & Xinqiang Qian & Panos M. Pardalos, 2022. "Inverse max+sum spanning tree problem under weighted $$l_{\infty }$$ l ∞ norm by modifying max-weight vector," Journal of Global Optimization, Springer, vol. 84(3), pages 715-738, November.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:3:d:10.1007_s10898-022-01170-y
    DOI: 10.1007/s10898-022-01170-y
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    References listed on IDEAS

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    1. Xianyue Li & Zhao Zhang & Ding-Zhu Du, 2018. "Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$ l p -norm," Journal of Global Optimization, Springer, vol. 70(3), pages 677-685, March.
    2. Xianyue Li & Zhao Zhang & Ruowang Yang & Heping Zhang & Ding-Zhu Du, 2020. "Approximation algorithms for capacitated partial inverse maximum spanning tree problem," Journal of Global Optimization, Springer, vol. 77(2), pages 319-340, June.
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