Combinatorial Path Planning for a System of Multiple Unmanned Vehicles

In this dissertation, the problem of planning the motion of m Unmanned Vehicles (UVs) (or simply vehicles) through n points in a plane is considered. A motion plan for a vehicle is given by the sequence of points and the corresponding angles at which each point must be visited by the vehicle. We require that each vehicle return to the same initial location(depot) at the same heading after visiting the points. The objective of the motion planning problem is to choose at most q(≤ m) UVs and find their motion plans so that all the points are visited and the total cost of the tours of the chosen vehicles is a minimum amongst all the possible choices of vehicles and their tours. This problem is a generalization of the wellknown Traveling Salesman Problem (TSP) in many ways: (1) each UV takes the role of salesman (2) motion constraints of the UVs play an important role in determining the cost of travel between any two locations; in fact, the cost of the travel between any two locations depends on direction of travel along with the heading at the origin and destination, and (3) there is an additional combinatorial complexity stemming from the need to partition the points to be visited by each UV and the set of UVs that must be employed by the mission. In this dissertation, a sub-optimal, two-step approach to motion planning is presented to solve this problem:(1) the combinatorial problem of choosing the vehicles and their associated tours is based on Euclidean distances between points and (2) once the sequence of points to be visited is specified, the heading at each point is determined based on a Dynamic Programming scheme. The solution to the first step is based on a generalization of Held-Karp’s method. We modify the Lagrangian heuristics for finding a close sub-optimal solution. In the later chapters of the dissertation, we relax the assumption that all vehicles are homogenous. The motivation of heterogenous variant of Multi-depot, Multiple Traveling Salesmen Problem (MDMTSP) derives form applications involving Unmanned Aerial Vehicles (UAVs) or ground robots requiring multiple vehicles with different capabilities to visit a set of locations

A (3/2+ε)(3/2 + \varepsilon)-Approximation for Multiple TSP with a Variable Number of Depots

One of the most studied extensions of the famous Traveling Salesperson Problem (TSP) is the {\sc Multiple TSP}: a set of $m\geq 1$ salespersons collectively traverses a set of $n$ cities by $m$ non-trivial tours, to minimize the total length of their tours. This problem can also be considered to be a variant of {\sc Uncapacitated Vehicle Routing} where the objective function is the sum of all tour lengths. When all $m$ tours start from a single common \emph{depot} $v_0$, then the metric {\sc Multiple TSP} can be approximated equally well as the standard metric TSP, as shown by Frieze (1983). The {\sc Multiple TSP} becomes significantly harder to approximate when there is a \emph{set} $D$ of $d \geq 1$ depots that form the starting and end points of the $m$ tours. For this case only a $(2-1/d)$-approximation in polynomial time is known, as well as a $3/2$-approximation for \emph{constant} $d$ which requires a prohibitive run time of $n^{\Theta(d)}$ (Xu and Rodrigues, \emph{INFORMS J. Comput.}, 2015). A recent work of Traub, Vygen and Zenklusen (STOC 2020) gives another approximation algorithm for {\sc Multiple TSP} running in time $n^{\Theta(d)}$ and reducing the problem to approximating TSP. In this paper we overcome the $n^{\Theta(d)}$ time barrier: we give the first efficient approximation algorithm for {\sc Multiple TSP} with a \emph{variable} number $d$ of depots that yields a better-than-2 approximation. Our algorithm runs in time $(1/\varepsilon)^{\mathcal O(d\log d)}\cdot n^{\mathcal O(1)}$, and produces a $(3/2+\varepsilon)$-approximation with constant probability. For the graphic case, we obtain a deterministic $3/2$-approximation in time $2^d\cdot n^{\mathcal O(1)}$.ithm for metric {\sc Multiple TSP} with run time $n^{\Theta(d)}$, which reduces the problem to approximating metric TSP.Comment: To be published at ESA 202

Combinatorial Path Planning for a System of Multiple Unmanned Vehicles

In this dissertation, the problem of planning the motion of m Unmanned Vehicles (UVs) (or simply vehicles) through n points in a plane is considered. A motion plan for a vehicle is given by the sequence of points and the corresponding angles at which each point must be visited by the vehicle. We require that each vehicle return to the same initial location(depot) at the same heading after visiting the points. The objective of the motion planning problem is to choose at most q(≤ m) UVs and find their motion plans so that all the points are visited and the total cost of the tours of the chosen vehicles is a minimum amongst all the possible choices of vehicles and their tours. This problem is a generalization of the wellknown Traveling Salesman Problem (TSP) in many ways: (1) each UV takes the role of salesman (2) motion constraints of the UVs play an important role in determining the cost of travel between any two locations; in fact, the cost of the travel between any two locations depends on direction of travel along with the heading at the origin and destination, and (3) there is an additional combinatorial complexity stemming from the need to partition the points to be visited by each UV and the set of UVs that must be employed by the mission. In this dissertation, a sub-optimal, two-step approach to motion planning is presented to solve this problem:(1) the combinatorial problem of choosing the vehicles and their associated tours is based on Euclidean distances between points and (2) once the sequence of points to be visited is specified, the heading at each point is determined based on a Dynamic Programming scheme. The solution to the first step is based on a generalization of Held-Karp’s method. We modify the Lagrangian heuristics for finding a close sub-optimal solution. In the later chapters of the dissertation, we relax the assumption that all vehicles are homogenous. The motivation of heterogenous variant of Multi-depot, Multiple Traveling Salesmen Problem (MDMTSP) derives form applications involving Unmanned Aerial Vehicles (UAVs) or ground robots requiring multiple vehicles with different capabilities to visit a set of locations

An Approach for Solving Multiple Travelling Salesman Problem using Ant Colony Optimization

Ant Colony Optimization (ACO) is a heuristic algorithm which has been proven a successful technique and applied to a number of combinatorial optimization (CO) problems. The traveling salesman problem (TSP) is one of the most important combinatorial problems. Multiple traveling salesman problem (MTSP) is a typical computationally complex combinatorialOptimization problem, which is an extension of the famous traveling salesman problem (TSP). The paper proposed an approach to show how the ant colony optimization (ACO) can be applied to the MTSP with ability constraint. There are several reasons for the choice of the TSP as the problem to explain the working of ACO algorithms: it is an important NP-hard optimization problem that arises in several applications; it is a problem to which ACO algorithms are easily applied; it is easily understandable, so that the algorithm behavior is not obscured by too many technicalities; and it is a standard test bed for new algorithmic ideas as a good performance on the TSP is often taken as a proof of their usefulness. Keywords— Ant colony optimization, Traveling salesman proble

Primal-Dual 2-Approximation Algorithm for the Monotonic Multiple Depot Heterogeneous Traveling Salesman Problem

We study a Multiple Depot Heterogeneous Traveling Salesman Problem (MDHTSP) where the cost of the traveling between any two targets depends on the type of the vehicle. The travel costs are assumed to be symmetric, satisfy the triangle inequality, and are monotonic, i.e., the travel costs between any two targets monotonically increases with the index of the vehicles. Exploiting the monotonic structure of the travel costs, we present a 2-approximation algorithm based on the primal-dual method

Routing Vehicles with Motion, Resource and Mission Constraints: Algorithms and Bounds

Unmanned Aerial Vehicles (UAVs) are used for several military and civil applications such as reconnaissance, surveillance etc. The UAVs, due to their design and size limitations, have inherent kinematic constraints, communication constraints etc. This thesis considers the path planning problems for UAVs while satisfying a class of constraints. We consider a multiple depot UAV routing problem, where the vehicles have motion constraints due to bound on their yaw-rate. For a given set of targets, it is required that each target should be on the path of at least one of the vehicles. This problem is hard to solve and currently there are no algorithm that could find an optimal solution. We aim to find tight lower bounds for this problem via Lagrangian relaxation. The complicating constraints of the problem are relaxed, and the cost function is penalized whenever those constraints are violated. This reduces the original problem to a known problem - a standard multiple traveling salesmen problem (MTSP). Simulation results are presented to show that this method significantly improved the existing lower bounds. The second problem we consider is the routing of UAVs in GPS denied environments and with limited communication range. Two different architectures for navigation assisted by an array of Unattended Ground Sensors (UGSs) are considered. In the first case, when an UAV localizes itself by communicating with an UGS, the second UAV can orbit around the first UAV. Contact with UGS allows them to act as beacons for relative navigation eliminating the need for GPS. A randomized algorithm with approximation ratio of 9/2 and a transformation technique are developed to solve this problem. In the second architecture, when two UAVs are located at two different UGSs, the third UAV localizes by triangulation using range measurements from the first two UAVs. This three UAV case is solved using a graph transformation technique to pose it as an one-in-a-set TSP. The solutions produced by these algorithms were used to simulate the UAV routing on AMASE, a simulation tool for routing UAVs developed by the Air Force Research Laboratories

Traveling Salesman Problem

This book is a collection of current research in the application of evolutionary algorithms and other optimal algorithms to solving the TSP problem. It brings together researchers with applications in Artificial Immune Systems, Genetic Algorithms, Neural Networks and Differential Evolution Algorithm. Hybrid systems, like Fuzzy Maps, Chaotic Maps and Parallelized TSP are also presented. Most importantly, this book presents both theoretical as well as practical applications of TSP, which will be a vital tool for researchers and graduate entry students in the field of applied Mathematics, Computing Science and Engineering

An exact algorithm for the vehicle routing problem with backhauls

Cataloged from PDF version of article.We consider the Vehicle Routing Problem with Backhauls, in which a eet of vehicles located at a central depot is to be used to serve a set of customers partitioned into two subsets of linehaul and backhaul customers. The ob jective of the problem is to minimize the total distance traveled by the entire eet. The problem is known to be N P-hard in the strongest sense and nds many practical applications in distribution planning. We present an exact algorithm for the Asymmetric Vehicle Routing Problem with Backhauls based on solving a relaxation of the problem. In a cutting plane fashion, the algorithm iteratively solves the relaxation while at each iteration, infeasible solutions are identi ed and seperated from the feasible set of the relaxation. The procedures to identify infeasible solutions are presented, and a set of cuts to eliminate these solutions is proposed. Local search procedures are incorporated to improve the algorithm. Computational tests on randomly generated instances, involving up to 90 customers, are given. The results show the e ectiveness of the proposed approach.Geloğulları, Cumhur AlperM.S