Abstract
This paper considers the problem of evacuating people located at vertices to a “sink” in a cycle network. In the “minmax-regret” version of this problem, the exact number of evacuees at each vertex is unknown, but only an interval for a possible number is given. We show that a minmax-regret 1-sink in cycle networks with uniform edge capacities can be found in \(O(n^2)\) time, where n is the number of vertices. No correct algorithm was known before for this problem.
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Notes
- 1.
We thank Prof. M. Golin of Hong Kong University of Science and Technology for pointing out that their claim is incorrect.
- 2.
Formally, they are the non-dominated scenarios defined in Sect. 2.4.
- 3.
From now on clockwise and counterclockwise are abbreviated as cw and ccw, respectively.
- 4.
In the parlance of network flow theory, flow obeying the latter condition is called confluent.
- 5.
We have \(\varTheta _{cw}(x,\hat{e})=0\) (resp. \(\varTheta ^s_{ccw}(x,\hat{e})=0\)), if x and \(\hat{e}\) are on the same edge, since \(V[v_{cw}(\hat{e}),x)=\emptyset \) (resp. \(V(x,v_{ccw}(\hat{e})]=\emptyset \)).
- 6.
The tiny circle at an end of each linear segment means that that point is missing.
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Acknowledgement
This work is supported in part by NSERC of Canada Discovery Grant, awarded to Robert Benkoczi and Binay Bhattacharya, in part by JST Crest (JPMJCR1402), granted to Naoki Kato and Yuya Higashikawa, and in part by JSPS Kakenhi Grant-in-Aid for Young Scientists (B) (17K12641), granted to Yuya Higashikawa.
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Benkoczi, R., Bhattacharya, B., Higashikawa, Y., Kameda, T., Katoh, N. (2019). Minmax-Regret Evacuation Planning for Cycle Networks. In: Gopal, T., Watada, J. (eds) Theory and Applications of Models of Computation. TAMC 2019. Lecture Notes in Computer Science(), vol 11436. Springer, Cham. https://doi.org/10.1007/978-3-030-14812-6_4
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