Abstract
We consider the bus evacuation problem. Given a positive integer B, a bipartite graph G with parts S and \(T \cup \{r\}\) in a metric space and functions \(l_i :S \rightarrow {\mathbb {Z}}_+\) and \({u_j :T \rightarrow \mathbb {Z}_+ \cup \{\infty \}}\), 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 \(l_i\) times and each vertex j of T must be visited at most \(u_j\) 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 \(\text{ NP }\)-hard and present a 10.2-approximation algorithm. Further, for the uncapacitated BEP, in which \(u_j\) is infinity for each j, we give a 4.2-approximation algorithm.
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Supported by Grant #2015/11937-9, São Paulo Research Foundation (FAPESP) and Grants #425340/2016-3, #308689/2017-8 and #313026/2017-3, National Council for Scientific and Technological Development (CNPq).
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Pedrosa, L.L.C., Schouery, R.C.S. Approximation algorithms for the bus evacuation problem. J Comb Optim 36, 131–141 (2018). https://doi.org/10.1007/s10878-018-0290-x
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DOI: https://doi.org/10.1007/s10878-018-0290-x