On the distance constrained vehicle routing problem
Operations Research
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Approximation algorithms
The vehicle routing problem
Decision-Aiding Methodology for the School Bus Routing and Scheduling Problem
Transportation Science
Approximation Algorithms for Orienteering and Discounted-Reward TSP
SIAM Journal on Computing
Improved algorithms for orienteering and related problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
PTAS for k-Tour Cover Problem on the Plane for Moderately Large Values of k
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
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The School Bus Problem is an NP-hard vehicle routing problem in which the goal is to route buses that transport children to a school such that for each child, the distance travelled on the bus does not exceed the shortest distance from the child's home to the school by more than a given regret threshold. Subject to this constraint and bus capacity limit, the goal is to minimize the number of buses required. In this paper, we give a polynomial time 4-approximation algorithm when the children and school are located at vertices of a fixed tree. As a byproduct of our analysis, we show that the integrality gap of the natural set-cover formulation for this problem is also bounded by 4. We also present a constant approximation for the variant where we have a fixed number of buses to use, and the goal is to minimize the maximum regret.