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Minmax regret 1-sink location problems on dynamic flow path networks with parametric weights

Author

Listed:
  • Tetsuya Fujie

    (University of Hyogo)

  • Yuya Higashikawa

    (University of Hyogo)

  • Naoki Katoh

    (University of Hyogo)

  • Junichi Teruyama

    (University of Hyogo)

  • Yuki Tokuni

    (University of Hyogo)

Abstract

This paper addresses the minmax regret 1-sink location problem on a dynamic flow path network with parametric weights. A dynamic flow path network consists of an undirected path with positive edge lengths, positive edge capacities, and nonnegative vertex weights. A path can be considered as a road, an edge length as the distance along the road, and a vertex weight as the number of people at the site. An edge capacity limits the number of people that can enter the edge per unit time. We consider the problem of locating a sink where all the people evacuate quickly. In our model, each weight is represented by a linear function of a common parameter t, and the decision maker who determines the sink location does not know the value of t. We formulate the problem under such uncertainty as the minmax regret problem. Given t and sink location x, the cost is the sum of arrival times at x for all the people determined by t. The regret for x under t is the gap between this cost and the optimal cost under t. The problem is to find the sink location minimizing the maximum regret over all t. For the problem, we propose an $$O(n^4 2^{\alpha (n)} \alpha (n)^2 \log n)$$ O ( n 4 2 α ( n ) α ( n ) 2 log n ) time algorithm, where n is the number of vertices in the network and $$\alpha (\cdot )$$ α ( · ) is the inverse Ackermann function. Also, for the special case in which every edge has the same capacity, we show that the complexity can be reduced to $$O(n^3 2^{\alpha (n)} \alpha (n) \log n)$$ O ( n 3 2 α ( n ) α ( n ) log n ) .

Suggested Citation

  • Tetsuya Fujie & Yuya Higashikawa & Naoki Katoh & Junichi Teruyama & Yuki Tokuni, 2024. "Minmax regret 1-sink location problems on dynamic flow path networks with parametric weights," Journal of Combinatorial Optimization, Springer, vol. 48(2), pages 1-20, September.
  • Handle: RePEc:spr:jcomop:v:48:y:2024:i:2:d:10.1007_s10878-024-01199-7
    DOI: 10.1007/s10878-024-01199-7
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    References listed on IDEAS

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    1. Li, Hongmei & Xu, Yinfeng, 2016. "Minimax regret 1-sink location problem with accessibility in dynamic general networks," European Journal of Operational Research, Elsevier, vol. 250(2), pages 360-366.
    2. Hongmei Li & Yinfeng Xu & Guanqun Ni, 2016. "Minimax regret vertex 2-sink location problem in dynamic path networks," Journal of Combinatorial Optimization, Springer, vol. 31(1), pages 79-94, January.
    3. L. R. Ford & D. R. Fulkerson, 1958. "Constructing Maximal Dynamic Flows from Static Flows," Operations Research, INFORMS, vol. 6(3), pages 419-433, June.
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