Traveling Salesman Problem Tour Research Articles

Optical solution for bounded NP-complete problems

We present a new optical method for solving bounded (input-length-restricted) NP-complete combinatorial problems. We have chosen to demonstrate the method with an NP-complete problem called the traveling salesman problem (TSP). The power of optics in this method is realized by using a fast matrix-vector multiplication between a binary matrix, representing all feasible TSP tours, and a gray-scale vector, representing the weights among the TSP cities. The multiplication is performed optically by using an optical correlator. To synthesize the initial binary matrix representing all feasible tours, an efficient algorithm is provided. Simulations and experimental results prove the validity of the new method.

Design of Steel Frames Using Ant Colony Optimization

A design procedure utilizing an ant colony optimization (ACO) technique is developed for discrete optimization of steel frames. The objective function considered is the total weight (or cost) of the structure subjected to serviceability and strength requirements as specified by the American Institute for Steel Construction (AISC) Load and Resistance Factor Design, 2001. The design of steel frames is mapped into a modified traveling salesman problem (TSP) where the configuration of the TSP network reflects the structural topology, and the resulting length of the TSP tour corresponds to the weight of the frame. The number of potential paths between nodes in the TSP network represents all (or a portion) of the available W-shapes in the AISC database. The resulting frame, mapped into a TSP, is minimized using an ACO algorithm with a penalty function to enforce strength and serviceability constraints. A comparison is presented between the ACO frame designs and designs developed using a genetic algorithm and classical continuous optimization methods.

Data structures and ejection chains for solving large-scale traveling salesman problems

Data structures play a crucial role in the efficient implementation of local search algorithms for problems that require circuit optimization in graphs. The traveling salesman problem (TSP) is the benchmark problem used in this study where two implementations of the stem-and-cycle (S&C) ejection chain algorithm are compared. The first implementation uses an Array data structure organized as a doubly linked list to represent TSP tours as well as the S&C reference structure. The second implementation considers a two-level tree structure. The motivation for this study comes from the fact that the S&C neighborhood structure usually requires subpaths to be reversed in order to preserve a feasible orientation for the resulting tour. The traditional Array structure proves to be inefficient for large-scale problems since to accomplish a path reversal it is necessary to update the predecessor and the successor of each node on the path to be reversed. Computational results performed on a set of benchmark problems up to 316,228 nodes clearly demonstrate the relative efficiency of the two-level tree data structure.

Design of Space Trusses Using Ant Colony Optimization

A design procedure utilizing an ant colony optimization (ACO) technique is developed for discrete optimization of space trusses. The objective function considered is the total weight (or cost) of the structure subjected to material and performance constraints in the form of stress and deflection limits. The design of space trusses using discrete variables is transformed into a modified traveling salesman problem (TSP) where the network of the TSP reflects the structural topology and the length of the TSP tour is the weight of the structure. The resulting truss, mapped into a TSP, is minimized using an ACO algorithm. The ACO design procedure uses discrete design variables, an open format for prescribing constraints, a penalty function to enforce design constraints, and allows for multiple loading cases. A comparison is presented between the ACO truss design procedure and designs developed using a genetic algorithm and classical continuous optimization methods.

On the capacitated vehicle routing problem

We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY.

The traveling salesman problem: new polynomial approximation algorithms and domination analysis

For the traveling salesman problem (TSP) on n nodes, we show that a tour which is better than ω((n/2)!) alternative tours can be identified in O(n 3) time. Also we provide another algorithm of complexity O n 3) which is guaranteed to produce a solution which is better than ω(n/(k+1))!(k!) n/(k+1) alternative tours, whenever k is fixed. Further we show that by using an O(n 4) algorithm, a TSP tour which dominates ω((n/2)!n) alternative tours can be identified. We also suggest an O(n 5) algorithm with improved domination number.