Drone Delivery Optimization

This research has addressed three critical challenges inherent in the implementation of drone delivery systems, namely, optimizing battery charging station placement, solving the shortest path problem for drones within their single battery charge travel distance, and efficiently scheduling multiple drones across numerous warehouses and delivery locations with diverse demands. The study has leveraged a 2D grid model with obstacles, providing a practical foundation extendable to a 3D grid for accommodating complex structures. For battery station placement, the Miller-Tucker-Zemlin subtour elimination method has been applied to avoid the formation of charging station clusters. Future research directions involve the integration of these cases into a holistic solution, exploration of three-dimensional space, and the pursuit of bi-level optimization considering the interdependence of battery station placement and shortest path determination. This study contributes to the emerging field of drone delivery systems by addressing key optimization challenges and paving the way for comprehensive, integrated solutions

Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $\epsilon > 0$, the algorithm outputs in $O(m \log^4n /\epsilon^2)$ time, with high probability, a $(1+\epsilon)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/\epsilon^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + \epsilon\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm