The Traveling Salesman Problem: Deceptivley Easy to State; Notoriously Hard to Solve

The purpose of this thesis is to give an overview of the history of the Traveling Salesman Problem and to show how it has been an integral part of the development of the fields of Integer Programming, and Combinatorial Optimization. The thesis starts in the 1800s and progresses through current attempts on solutions of the problem. The thesis is not meant to describe in detail every attempt made, nor to describe an original solution, but to provide a high level overview of every solution attempt, and to guide the reader on what has been done, and what still can be done

Determine the Optimal Sequence-Dependent Completion Times for Multiple Demand with Multi-Products Using Genetic Algorithm

Sequencing is the most impact factor on the total completion time , the products sequences inside demands that consist from muti-product and for multiple demands . It is very important in assembly line and batch production . The most important drawback of existing methods used to solve the sequencing problems is the sequence must has a few products and dependent completion time for single demand . In this paper we used genetic algorithm –based Travelling Salesman Problem with Precedence Constraints Approach ( TSPPCA)  to minimize completion time . The main advantage of this new method , it is used to solve the sequencing problems for multiple demand with multi-product In this paper , we compare between modify the assignment method ( MAM ) and genetic algorithm  depend on least completion time , the results discern that   GA  has minimum completion time Keywords: products sequences , completion time , travel salesman problem (TSP ) , TSPPCA , genetic algorithm. 

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

Polyhedral techniques in combinatorial optimization

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Provably good solutions for the traveling salesman problem

The determination of true optimum solutions of combinatorial optimization problems is seldomly required in practical applications. The majority of users of optimization software would be satisfied with solutions of guaranteed quality in the sense that it can be proven that the given solution is at most a few percent off an optimum solution. This paper presents a general framework for practical problem solving with emphasis on this aspect. A detailed discussion along with a report about extensive computational experiments is given for the traveling salesman problem

Application and Assessment of Divide-and-Conquer-based Heuristic Algorithms for some Integer Optimization Problems

In this paper three heuristic algorithms using the Divide-and-Conquer paradigm are developed and assessed for three integer optimizations problems: Multidimensional Knapsack Problem (d-KP), Bin Packing Problem (BPP) and Travelling Salesman Problem (TSP). For each case, the algorithm is introduced, together with the design of numerical experiments, in order to empirically establish its performance from both points of view: its computational time and its numerical accuracy.Comment: 16 pages, 6 figures, 8 table