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

Approximation Algorithms for Union and Intersection Covering Problems

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here nodes and edges represent requests and items, respectively. In this paper, we initiate the study of a new family of multi-layer covering problems. Each such problem consists of a collection of h distinct instances of a standard covering problem (layers), with the constraint that all layers share the same set of requests. We identify two main subfamilies of these problems: - in a union multi-layer problem, a request is satisfied if it is satisfied in at least one layer; - in an intersection multi-layer problem, a request is satisfied if it is satisfied in all layers. To see some natural applications, consider both generalizations of k-MST. Union k-MST can model a problem where we are asked to connect a set of users to at least one of two communication networks, e.g., a wireless and a wired network. On the other hand, intersection k-MST can formalize the problem of connecting a subset of users to both electricity and water. We present a number of hardness and approximation results for union and intersection versions of several standard optimization problems: MST, Steiner tree, set cover, facility location, TSP, and their partial covering variants

Towards the solution of variants of Vehicle Routing Problem

Some of the problems that are used extensively in -real life are NP complete problems. There is no any algorithm which can give the optimal solution to NP complete problems in the polynomial time in the worst case. So researchers are applying their best efforts to design the approximation algorithms for these NP complete problems. Approximation algorithm gives the solution of a particular problem, which is close to the optimal solution of that problem. In this paper, a study on variants of vehicle routing problem is being done along with the difference in the approximation ratios of different approximation algorithms as being given by researchers and it is found that Researchers are continuously applying their best efforts to design new approximation algorithms which have better approximation ratio as compared to the previously existing algorithms

Quantum-Assisted Solution Paths for the Capacitated Vehicle Routing Problem

Many relevant problems in industrial settings result in NP-hard optimization problems, such as the Capacitated Vehicle Routing Problem (CVRP) or its reduced variant, the Travelling Salesperson Problem (TSP). Even with today's most powerful classical algorithms, the CVRP is challenging to solve classically. Quantum computing may offer a way to improve the time to solution, although the question remains open as to whether Noisy Intermediate-Scale Quantum (NISQ) devices can achieve a practical advantage compared to classical heuristics. The most prominent algorithms proposed to solve combinatorial optimization problems in the NISQ era are the Quantum Approximate Optimization Algorithm (QAOA) and the more general Variational Quantum Eigensolver (VQE). However, implementing them in a way that reliably provides high-quality solutions is challenging, even for toy examples. In this work, we discuss decomposition and formulation aspects of the CVRP and propose an application-driven way to measure solution quality. Considering current hardware constraints, we reduce the CVRP to a clustering phase and a set of TSPs. For the TSP, we extensively test both QAOA and VQE and investigate the influence of various hyperparameters, such as the classical optimizer choice and strength of constraint penalization. Results of QAOA are generally of limited quality because the algorithm does not reach the energy threshold for feasible TSP solutions, even when considering various extensions such as recursive, warm-start and constraint-preserving mixer QAOA. On the other hand, the VQE reaches the energy threshold and shows a better performance. Our work outlines the obstacles to quantum-assisted solutions for real-world optimization problems and proposes perspectives on how to overcome them.Comment: Submitted to the IEEE for possible publicatio

A Swarm of Salesmen: Algorithmic Approaches to Multiagent Modeling

This honors thesis describes the algorithmic abstraction of a problem modeling a swarm of Mars rovers, where many agents must together achieve a goal. The algorithmic formulation of this problem is based on the traveling salesman problem (TSP), and so in this thesis I offer a review of the mathematical technique of linear programming in the context of its application to the TSP, an overview of some variations of the TSP and algorithms for approximating and solving them, and formulations without solutions of two novel TSP variations which are useful for modeling the original problem