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Sink Evacuation on Trees with Dynamic Confluent Flows

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Di Chen and Mordecai Golin. Sink Evacuation on Trees with Dynamic Confluent Flows. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ISAAC.2016.25

Abstract

Let G = (V, E) be a graph modelling a building or road network in which edges have-both travel times (lengths) and capacities associated with them. An edge’s capacity is the number of people that can enter that edge in a unit of time. In emergencies, people evacuate towards the exits. If too many people try to evacuate through the same edge, congestion builds up and slows down the evacuation. Graphs with both lengths and capacities are known as Dynamic Flow networks. An evacuation plan for G consists of a choice of exit locations and a partition of the people at the vertices into groups, with each group evacuating to the same exit. The evacuation time of a plan is the time it takes until the last person evacuates. The k-sink evacuation problem is to provide an evacuation plan with k exit locations that minimizes the evacuation time. It is known that this problem is NP-Hard for general graphs but no polynomial time algorithm was previously known even for the case of G a tree. This paper presents an O(nk^2 log^5 n) algorithm for the k-sink evacuation problem on trees, which can also be applied to a more general class of problems.
Keywords
• Sink Evacuation
• Dynamic Flow
• Facility Location
• Parametric Search

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References

1. J. E. Aronson. A survey of dynamic network flows. Annals of Operations Research, 20(1):1-66, 1989. URL: http://dx.doi.org/10.1007/BF02216922.
2. G. P. Arumugam, J. Augustine, M. J. Golin, and P. Srikanthan. Optimal evacuation on dynamic paths with general capacities of edges. Unpublished Manuscript, 2015.
3. J. Chen, R. Rajaraman, and R. Sundaram. Meet and merge: Approximation algorithms for confluent flows. Journal of Computer and System Sciences, 72(3):468-489, 2006.
4. Jiangzhuo Chen, Robert D berg, László Lovász, Rajmohan Rajaraman, Ravi Sundaram, and Adrian Vetta. (Almost) Tight bounds and existence theorems for single-commodity confluent flows. Journal of the ACM, 54(4), jul 2007.
5. Daniel Dressler and Martin Strehler. Capacitated Confluent Flows: Complexity and Algorithms. In 7th International Conference on Algorithms and Complexity (CIAC'10), pages 347-358, 2010.
6. Lisa Fleischer and Martin Skutella. Quickest Flows Over Time. SIAM Journal on Computing, 36(6):1600-1630, January 2007.
7. L. R. Ford and D. R. Fulkerson. Constructing Maximal Dynamic Flows from Static Flows. Operations Research, 6(3):419-433, June 1958.
8. Greg N Frederickson. Parametric search and locating supply centers in trees. In Proceedings of the Second Workshop on Algorithms and Data Structures (WADS'91), pages 299-319. Springer, 1991.
9. Michael R. Garey and David S. Johnson. Computers and intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979.
10. Yuya Higashikawa, M. J. Golin, and Naoki Katoh. Minimax Regret Sink Location Problem in Dynamic Tree Networks with Uniform Capacity. In The 8th International Workshop on Algorithms and Computation (WALCOM'2014), pages 125-137, 2014. URL: http://dx.doi.org/10.1007/978-3-319-04657-0_14.
11. Yuya Higashikawa, Mordecai J Golin, and Naoki Katoh. Multiple sink location problems in dynamic path networks. Theoretical Computer Science, 607:2-15, 2015.
12. B Hoppe and É Tardos. The quickest transshipment problem. Mathematics of Operations Research, 25(1):36-62, 2000.
13. N. Kamiyama, N. Katoh, and A. Takizawa. Theoretical and Practical Issues of Evacuation Planning in Urban Areas. In The Eighth Hellenic European Research on Computer Mathematics and its Applications Conference (HERCMA2007), pages 49-50, 2007.
14. Satoko Mamada and Kazuhisa Makino. An Evacuation Problem in Tree Dynamic Networks with Multiple Exits. In Tatsuo Arai, Shigeru Yamamoto, and Kazuhi Makino, editors, Systems &Human Science-For Safety, Security, and Dependability; Selected Papers of the 1st International Symposium SSR2003, pages 517-526. Elsevier B.V, 2005.
15. Satoko Mamada, Takeaki Uno, Kazuhisa Makino, and Satoru Fujishige. A tree partitioning problem arising from an evacuation problem in tree dynamic networks. Journal of the Operations Research Society of Japan, 48(3):196-206, 2005.
16. Satoko Mamada, Takeaki Uno, Kazuhisa Makino, and Satoru Fujishige. An O(n log² n)algorithm for the optimal sink location problem in dynamic tree networks. Discrete Applied Mathematics, 154(2387-2401):251-264, 2006.
17. M. M. B. Pascoal, M. Eugénia V. Captivo, and J. C. N. Clímaco. A comprehensive survey on the quickest path problem. Annals of Operations Research, 147(1):5-21, August 2006.
18. F. Bruce Shepherd and Adrian Vetta. The Inapproximability of Maximum Single-Sink Unsplittable, Priority and Confluent Flow Problems, 2015. URL: http://arxiv.org/abs/1504.0627.
19. Martin Skutella. An introduction to network flows over time. In William Cook, László Lovász, and Jens Vygen, editors, Research Trends in Combinatorial Optimization, pages 451-482. Springer, 2009.
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