How to collect balls moving in the Euclidean plane

AbstractIn this paper, we study how to collect n balls moving with a fixed constant velocity in the Euclidean plane by k robots moving on straight track-lines through the origin. Since all the balls might not be caught by robots, differently from Moving-target TSP, we consider the following 3 problems in various situations: (i) deciding if k robots can collect all n balls; (ii) maximizing the number of the balls collected by k robots; (iii) minimizing the number of the robots to collect all n balls. The situations considered in this paper contain the cases in which track-lines are given (or not), and track-lines are identical (or not). For all problems and situations, we provide polynomial time algorithms or proofs of intractability, which clarify the tractability–intractability frontier in the ball collecting problems in the Euclidean plane

A Systematic Review of Approximability Results for Traveling Salesman Problems leveraging the TSP-T3CO Definition Scheme

The traveling salesman (or salesperson) problem, short TSP, is a problem of strong interest to many researchers from mathematics, economics, and computer science. Manifold TSP variants occur in nearly every scientific field and application domain: engineering, physics, biology, life sciences, and manufacturing just to name a few. Several thousand papers are published on theoretical research or application-oriented results each year. This paper provides the first systematic survey on the best currently known approximability and inapproximability results for well-known TSP variants such as the "standard" TSP, Path TSP, Bottleneck TSP, Maximum Scatter TSP, Generalized TSP, Clustered TSP, Traveling Purchaser Problem, Profitable Tour Problem, Quota TSP, Prize-Collecting TSP, Orienteering Problem, Time-dependent TSP, TSP with Time Windows, and the Orienteering Problem with Time Windows. The foundation of our survey is the definition scheme T3CO, which we propose as a uniform, easy-to-use and extensible means for the formal and precise definition of TSP variants. Applying T3CO to formally define the variant studied by a paper reveals subtle differences within the same named variant and also brings out the differences between the variants more clearly. We achieve the first comprehensive, concise, and compact representation of approximability results by using T3CO definitions. This makes it easier to understand the approximability landscape and the assumptions under which certain results hold. Open gaps become more evident and results can be compared more easily

Assessment of genetic algorithm based assignment strategies for unmanned systems using the multiple traveling salesman problem with moving targets

Title from PDF of title page, viewed March 1, 2023Thesis advisor: Travis FieldsVitaIncludes bibliographical references (pages 100-106)Thesis (M.S.)--Department of Civil and Mechanical Engineering. University of Missouri--Kansas City, 2021The continuous and rapid advancements in autonomous unmanned systems technologies presents increasingly sophisticated threats to military operations. These threats necessitate the prioritization of improved strategies for military resources and air base defense. In many scenarios, it is necessary to combat hostile unmanned systems before they reach the defensible perimeters of existing fixed-base defense systems. One solution to this problem is weaponizing friendly unmanned systems to hunt and kill hostile unmanned systems. However, the assignment and path planning of these “Hunter-Killer” systems to incoming hostile unmanned systems, in a multiple friendly versus multiple enemy scenario, presents a major challenge and can be represented by the Multiple Traveling Salesmen Problem with Moving Targets (MTSPMT). The MTSPMT is a combinatorial optimization problem and an extension of the classical Traveling Salesman Problem whereby the number of salesmen is increased and targets (cities) move with respect to time. The objective of the MTSPMT, for the application of military defense using a squadron of Hunter-Killer unmanned systems, is to determine a path that minimizes the cost required for multiple Hunter-Killer unmanned systems to successfully intercept all incoming threats. In this study, an assessment of genetic algorithm based assignment strategies for unmanned systems using the MTSPMT is performed. A number of scenarios were constructed using up to 50 hostile unmanned systems and the generated solutions were compared based on their resulting time to converge, solution fitness, and number of generations required. Findings indicate that under certain conditions genetic based algorithms provide better results on average and converge more rapidly than brute force searching and existing assignment and path planning solutions.Introduction -- Literature review -- Problem formulation and model design -- Methodology -- Results and Discussion -- Conclusio

A Guided Neighborhood Search Applied to the Split Delivery Vehicle Routing Problem

The classic vehicle routing problem considers the distribution of goods to geographically scattered customers from a central depot using a homogeneous fleet of vehicles with finite capacity. Each customer has a known demand and can be visited by exactly one vehicle. Each vehicle services the assigned customers in such a way that all customers are fully supplied and the total service does not exceed the vehicle capacity. In the split delivery vehicle routing problem, a customer can be visited by more than one vehicle, i.e., a customer demand can be split between various vehicles. Allowing split deliveries has been proven to potentially reduce the operational costs of the fleet. This study efficiently solves the split delivery vehicle routing problem using three new approaches. In the first approach, the problem is solved in two stages. During the first stage, an initial solution is found by means of a greedy approach that can produce high quality solutions comparable to those obtained with existing sophisticated approaches. The greedy approach is based on a novel concept called the route angle control measure that helps to produce spatially thin routes and avoids crossing routes. In the second stage, this constructive approach is extended to an iterative approach using adaptive memory concepts, and then a variable neighborhood descent process is added to improve the solution obtained. A new solution diversification scheme is presented in the second approach based on concentric rings centered at the depot that partitions the original problem. The resulting sub-problems are then solved using the greedy approach with route angle control measures. Different ring settings produce varied partitions and thus different solutions to the original problem are obtained and improved via a variable neighborhood descent. The third approach is a learning procedure based on a set or population of solutions. Those solutions are used to find attractive attributes and construct new solutions within a tabu search framework. As the search progresses, the existing population evolves, better solutions are included in it whereas bad solutions are removed from it. The initial set is constructed using the greedy approach with the route angle control measure whereas new solutions are created using an adaptation of the well known savings algorithm of Clarke and Wright (1964) and improved by means of an enhanced version of the variable neighborhood descent process. The proposed approaches are tested on benchmark instances and results are compared with existing implementations

Moving-Target TSP and Related Problems

. Previous literature on the Traveling Salesman Problem (TSP) implicitly assumed that the sites to be visited are stationary. Motivated by practical applications, we introduce a time-dependent generalization of TSP which we call Moving-Target TSP, where a pursuer must intercept a set of targets which move with constant velocities in a minimum amount of time. We propose approximate and exact algorithms for several natural variants of Moving-Target TSP. Our implementation of the exact algorithm of One-Dimensional TSP is available on the Web. 1 Introduction The classical Traveling Salesman Problem (TSP) has been studied extensively, and many TSP heuristics have been proposed over the years (see surveys such as [4] [7]). Previous works on TSP have assumed that the cities/targets to be visited are stationary. However, several practical scenarios give rise to TSP instances where the targets to be visited are themselves in motion (e.g., when a supply ship resupplies patrolling boats, or when..