Approximation algorithms for the bus evacuation problem

Approximation algorithms for the bus evacuation problem We consider the bus evacuation problem. Given a positive integer B, a bipartite graph G with parts S and $$T \cup \{r\}$$ T ∪ { r } in a metric space and functions $$l_i :S \rightarrow {\mathbb {Z}}_+$$ l i : S → Z + and $${u_j :T \rightarrow \mathbb {Z}_+ \cup \{\infty \}}$$ u j : T → Z + ∪ { ∞ } , one wishes to find a set of B walks in G. Every walk in B should start at r and finish in T and r must be visited only once. Also, among all walks, each vertex i of S must be visited at least $$l_i$$ l i times and each vertex j of T must be visited at most  $$u_j$$ u j times. The objective is to find a solution that minimizes the length of the longest walk. This problem arises in emergency planning situations where the walks correspond to the routes of B buses that must transport each group of people in S to a shelter in T, and the objective is to evacuate the entire population in the minimum amount of time. In this paper, we prove that approximating this problem by less than a constant is $$\text{ NP }$$ NP -hard and present a 10.2-approximation algorithm. Further, for the uncapacitated BEP, in which $$u_j$$ u j is infinity for each j, we give a 4.2-approximation algorithm. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Combinatorial Optimization Springer Journals

Approximation algorithms for the bus evacuation problem

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operations Research/Decision Theory
ISSN
1382-6905
eISSN
1573-2886
D.O.I.
10.1007/s10878-018-0290-x
Publisher site
See Article on Publisher Site

Abstract

We consider the bus evacuation problem. Given a positive integer B, a bipartite graph G with parts S and $$T \cup \{r\}$$ T ∪ { r } in a metric space and functions $$l_i :S \rightarrow {\mathbb {Z}}_+$$ l i : S → Z + and $${u_j :T \rightarrow \mathbb {Z}_+ \cup \{\infty \}}$$ u j : T → Z + ∪ { ∞ } , one wishes to find a set of B walks in G. Every walk in B should start at r and finish in T and r must be visited only once. Also, among all walks, each vertex i of S must be visited at least $$l_i$$ l i times and each vertex j of T must be visited at most  $$u_j$$ u j times. The objective is to find a solution that minimizes the length of the longest walk. This problem arises in emergency planning situations where the walks correspond to the routes of B buses that must transport each group of people in S to a shelter in T, and the objective is to evacuate the entire population in the minimum amount of time. In this paper, we prove that approximating this problem by less than a constant is $$\text{ NP }$$ NP -hard and present a 10.2-approximation algorithm. Further, for the uncapacitated BEP, in which $$u_j$$ u j is infinity for each j, we give a 4.2-approximation algorithm.

Journal

Journal of Combinatorial OptimizationSpringer Journals

Published: Apr 17, 2018

References

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