Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem . Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O (log n / log log n ) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.

New graph algorithms via polyhedral techniques

This article gives a short overview of my dissertation, where new algorithms are given for two fundamental graph problems. We develop novel ways of using linear programming formulations, even exponential-sized ones, to extract structure from problem instances and to guide algorithms in making progress. The first part of the dissertation addresses a benchmark problem in combinatorial optimization: the asymmetric traveling salesman problem (ATSP). It consists in finding the shortest tour that visits all vertices of a given edge-weighted directed graph. A ρ-approximation algorithm for ATSP is one that runs in polynomial time and always produces a tour at most ρ times longer than the shortest tour. Finding such an algorithm with constant ρ had been a long-standing open problem. Here we give such an algorithm. The second part of the dissertation addresses the perfect matching problem. We have known since the 1980s that it has efficient parallel algorithms if the use of randomness is allowed. However, we do not know if randomness is necessary – that is, whether the matching problem is in the class NC. We show that it is in the class quasi-NC. That is, we give a deterministic parallel algorithm that runs in poly-logarithmic time on quasi-polynomially many processors.

Comparison of the Sub-Tour Elimination Methods for the Asymmetric Traveling Salesman Problem Applying the SECA Method

There are many sub-tour elimination constraint (SEC) formulations for the traveling salesman problem (TSP). Among the different methods found in articles, usually three apply more than others. This study examines the Danzig–Fulkerson–Johnson (DFJ), Miller–Tucker–Zemlin (MTZ), and Gavish–Graves (GG) formulations to select the best asymmetric traveling salesman problem (ATSP) formulation. The study introduces five criteria as the number of constraints, number of variables, type of variables, time of solving, and differences between the optimum and the relaxed value for comparing these constraints. The reason for selecting these criteria is that they have the most significant impact on the mathematical problem-solving complexity. A new and well-known multiple-criteria decision making (MCDM) method, the simultaneous evaluation of the criteria and alternatives (SECA) method was applied to analyze these criteria. To use the SECA method for ranking the alternatives and extracting information about the criteria from constraints needs computational computing. In this research, we use CPLEX 12.8 software to compute the criteria value and LINGO 11 software to solve the SECA method. Finally, we conclude that the Gavish–Graves (GG) formulation is the best. The new web-based software was used for testing the results.

Research of Features of the Combined Algorithm for Solving the Asymmetric Traveling Salesman Problem

The exact algorithm that implements the Branch and Boimd method with precomputed tour which is calculated by Lin-Kernighan-Helsgaun metaheuristic algorithm for solving the Traveling Salesman Problem is concerned here. Reducing the number of decision tree nodes, which are created by the Branches and Bound method, due to a "good" precomputed tour leads to the classical balancing dilemma of time costs. A tour that is close to optimal one takes time, even when the Lin-Kernighan-Helsgaun algorithm is used, however it reduces the working time of the Branch and Bound method. The problem of determining the scope of such a combined algorithm arises. In this article it is solved by using a special characteristic of the individual Traveling Salesman Problem — the number of changes tracing direction in the search decision tree generated by the Branch and Bound Method. The use of this characteristic allowed to divide individual tasks into three categories, for which, based on experimental data, recommendations of the combined algorithm usage are formulated. Based on the data obtained in a computational experiment (in range from 30 to 45), it is recommended to use a combined algorithm for category III problems starting with n = 36, and for category II problems starting with n = 42.

The Pheromone-Based Harmony Search Algorithm for the Asymmetric Traveling Salesman Problem

This paper presents a modification of the Harmony Search algorithm (HS) adjusted to an effective solving of instances of the Asymmetric Traveling Salesman Problem. The improvement of the technique spans the application of a pheromone, which, by serving the role of long-term memory, enables the improvement of the quality of determined results, especially for tasks characterized by a significant number of vertices. The publication includes the results of tests that suggest the achievement of effectiveness improvement through the modification of the HS and recommendations concerning the proper configuration of the algorithm.