Dynamic Programming is an optimization technique used in computer science and mathematics. Introduced in the 1950s, it has been applied to many classic combinatorial optimization problems, such as the Shortest Path Problem, the Knapsack Problem, and the Traveling Salesman Problem, with varying degrees of practical success. In this thesis, we present two applications of dynamic programming to optimization problems. The first application is as a method to compute the Branch-Cut-and-Price (BCP) family of lower bounds for the Traveling Salesman Problem (TSP), and several vehicle routing problems that generalize it. We then prove that the BCP family provides a set of lower bounds that is at least as strong as the Approximate Linear Program (ALP) family of lower bounds for the TSP. The second application is a novel dynamic programming model used to determine the placement of cuts for a particular form of skull surgery called Cranial Vault Remodeling.

University of Macau / Faculty of Science and Technology / Department of Mathematics

One of the simplest, but still NP-hard, routing problems is the Traveling Salesman Problem (TSP). In the TSP, one is given a set of cities and a way of measuring the distance between cities. One has to find the shortest tour that visits all cities exactly once and returns back to the starting city. In state-of-the-art algorithms, they all assume that a complete graph is given as an input. However, for very large graphs, generating all edges in a complete graph, which corresponds to finding shortest paths for all city pairs, could be time-consuming. This is definitely a major obstacle for some real-life applications, especially when the tour needs to be generated in real-time. The objective, in this thesis, is to find a near-optimal TSP tour with a reduced set of edges in the complete graph. In particular, the following problems are investigated: which subset of edges can be produced in a shorter time comparing to the time for generating the complete graph? Is there a subset of edges in the complete graph that results in a better near-optimal tour than other sets? With a non-complete graph, which improvement algorithms work better? In this thesis, we study six algorithms to generate subsets of edges in a complete graph. To evaluate the proposed algorithms, extensive experiments are conducted with the well-known TSP data in a TSP library. In these experiments, we evaluate these algorithms in terms of tour quality, time and scalability.

This thesis is a survey on the approximability of the asymmetric traveling salesmanproblem with triangle inequality (ATSP).In the ATSP we are given a set of cities and a function that gives the cost of travelingbetween any pair of cities. The cost function must satisfy the triangle inequality, i.e.the cost of traveling from city A to city B cannot be larger than the cost of travelingfrom A to some other city C and then to B. However, we allow the cost function tobe asymmetric, i.e. the cost of traveling from city A to city B may not equal the costof traveling from B to A. The problem is then to find the cheapest tour that visit eachcity exactly once. This problem is NP-hard, and thus we are mainly interested in approximationalgorithms. We study the repeated cycle cover heuristic by Frieze et al. We alsostudy the Held-Karp heuristic, including the recent result by Asadpour et al. that givesa new upper bound on the integrality gap. Finally we present the result ofPapadimitriou and Vempala which shows that it is NP-hard to approximate the ATSP with a ratio better than 117/116.

One of the simplest, but still NP-hard, routing problems is the Traveling Salesman Problem (TSP). In the TSP, one is given a set of cities and a way of measuring the distance between cities. One has to find the shortest tour that visits all cities exactly once and returns back to the starting city. In state-of-the-art algorithms, they all assume that a complete graph is given as an input. However, for very large graphs, generating all edges in a complete graph, which corresponds to finding shortest paths for all city pairs, could be time-consuming. This is definitely a major obstacle for some real-life applications, especially when the tour needs to be generated in real-time. The objective, in this thesis, is to find a near-optimal TSP tour with a reduced set of edges in the complete graph. In particular, the following problems are investigated: which subset of edges can be produced in a shorter time comparing to the time for generating the complete graph? Is there a subset of edges in the complete graph that results in a better near-optimal tour than other sets? With a non-complete graph, which improvement algorithms work better? In this thesis, we study six algorithms to generate subsets of edges in a complete graph. To evaluate the proposed algorithms, extensive experiments are conducted with the well-known TSP data in a TSP library. In these experiments, we evaluate these algorithms in terms of tour quality, time and scalability.

Previous research on DNA computing has shown that DNA algorithms are useful to solve some combinatorial problems, such as the Hamiltonian path problem and the traveling salesman problem. The basic concept implicit in previous DNA algorithms is the brute force method. That is, all possible solutions are created initially, then inappropriate solutions are eliminated, and finally the remaining solutions are correct or the best ones. However, correct solutions may be destroyed while the procedure is executed. In order to avoid such an error, we recommend combining the conventional concepts of DNA computing with a heuristic optimization method and apply the new approach to design strategies. In this thesis, we present a DNA algorithm based on ant colony optimization (ACO) for solving the traveling salesman problem (TSP). Our method manipulates DNA strands of candidate solutions initially. Even if the correct solutions are destroyed during the process of filtering out, the remaining solutions can be reconstructed and correct solutions can be reformed. After filtering out inappropriate solutions, we employ control of melting temperature to amplify the surviving DNA strings proportionally. The product is used as the input and the iteration is performed repeatedly. Accordingly, the concentration of correct solutions will be increased. Our results agree with that obtained by conventional ant colony optimization algorithms and are better than that obtained by genetic algorithms. The same idea can be applied to design methods for solving other combinatorial problems with DNA computing.

In this thesis we use greedy randomize adaptive search procedure (GRASP) to solve the traveling salesman problem (TSP). Starting with nearest neighbor method to construct the initial TSP tour, we apply the 2-opt and the path-relinking method for the initial tour improvement. To increase 2-opt search speed, fixed-radius near neighbor search and don0t − look bit techniques are introduced. For the same reason a new efficient data structure, the reverse array, is proposed to represent the TSP tour. Computational results show that GRASP gives fairly good solutions in a short time.

The ant system is a new meta-heuristic for hard combinatorial optimization problems. It is a population-based approach that uses exploitation of positive feedback as well as greedy search. It was first proposed for tackling the well known Traveling Salesman Problem (TSP), but has been also successfully applied to problems such as quadratic assignment, job-shop scheduling, vehicle routing and graph coloring.In this paper we introduce a new rank based version of the ant system and present results of a computational study, where we compare the ant system with simulated annealing and a genetic algorithm on several TSP instances. It turns out that our rank based ant system can compete with the other methods in terms of average behavior, and shows even better worst case behavior. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

Dynamic Programming is an optimization technique used in computer science and mathematics. Introduced in the 1950s, it has been applied to many classic combinatorial optimization problems, such as the Shortest Path Problem, the Knapsack Problem, and the Traveling Salesman Problem, with varying degrees of practical success. In this thesis, we present two applications of dynamic programming to optimization problems. The first application is as a method to compute the Branch-Cut-and-Price (BCP) family of lower bounds for the Traveling Salesman Problem (TSP), and several vehicle routing problems that generalize it. We then prove that the BCP family provides a set of lower bounds that is at least as strong as the Approximate Linear Program (ALP) family of lower bounds for the TSP. The second application is a novel dynamic programming model used to determine the placement of cuts for a particular form of skull surgery called Cranial Vault Remodeling.

Thesis (M.S.)--Cleveland State University, 2009. / Abstract. Title from PDF t.p. (viewed on Sept. 8, 2009). Includes bibliographical references (p. 70-73). Available online via the OhioLINK ETD Center and also available in print.