A note on the relationship between the Graphical Traveling Salesman Polyhedron, the Symmetric Traveling Salesman Polytope, and the Metric Cone

In this short communication, we observe that the Graphical Traveling Salesman Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric Traveling Salesman Polytope and the polar of the metric cone. This follows almost trivially from known facts. There are two reasons why we find this observation worth communicating none-the-less: It is very surprising; it helps to understand the relationship between these two important families of polyhedra.Comment: short communication (3 pages), Discrete Appl. Mat

TSP--Infrastructure for the Traveling Salesperson Problem

The traveling salesperson (or, salesman) problem (TSP) is a well known and important combinatorial optimization problem. The goal is to find the shortest tour that visits each city in a given list exactly once and then returns to the starting city. Despite this simple problem statement, solving the TSP is difficult since it belongs to the class of NP-complete problems. The importance of the TSP arises besides from its theoretical appeal from the variety of its applications. Typical applications in operations research include vehicle routing, computer wiring, cutting wallpaper and job sequencing. The main application in statistics is combinatorial data analysis, e.g., reordering rows and columns of data matrices or identifying clusters. In this paper, we introduce the R package TSP which provides a basic infrastructure for handling and solving the traveling salesperson problem. The package features S3 classes for specifying a TSP and its (possibly optimal) solution as well as several heuristics to find good solutions. In addition, it provides an interface to Concorde, one of the best exact TSP solvers currently available.

A Traveling Salesman\u27s Approach to Clustering Gene Expression Data

Given a matrix of values, rearrangement clustering involves rearranging the rows of the matrix and identifying cluster boundaries within the linear ordering of the rows. The TSP+k algorithm for rear-rangement clustering was presented in [3] and its implementation is described in this note. Using this code, we solve a 2,467-gene expression data clustering problem and identify “good” clusters that con-tain close to eight times the number of genes that were clustered by Eisen et al. (1998). Furthermore, we identify 106 functional groups that were overlooked in that paper. We make our implementation available to the general public for applications of gene expression data analysis

Speeding up IP-based Algorithms for Constrained Quadratic 0-1 Optimization

In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP).Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming.Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvatal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastical speedup in the solution of constrained quadratic 0-1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions

Generating partitions of a graph into a fixed number of minimum weight cuts

AbstractIn this paper, we present an algorithm for the generation of all partitions of a graph G with positive edge weights into k mincuts. The algorithm is an enumeration procedure based on the cactus representation of the mincuts of G. We report computational results demonstrating the efficiency of the algorithm in practice and describe in more detail a specific application for generating cuts in branch-and-cut algorithms for the traveling salesman problem

Speeding up IP-based Algorithms for Constrained Quadratic 0-1 Optimization

In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP).Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming.Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvatal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastical speedup in the solution of constrained quadratic 0-1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions

Constraint Programming Algorithms for Route Planning Exploiting Geometrical Information

Problems affecting the transport of people or goods are plentiful in industry and commerce and they also appear to be at the origin of much more complex problems. In recent years, the logistics and transport sector keeps growing supported by technological progress, i.e. companies to be competitive are resorting to innovative technologies aimed at efficiency and effectiveness. This is why companies are increasingly using technologies such as Artificial Intelligence (AI), Blockchain and Internet of Things (IoT). Artificial intelligence, in particular, is often used to solve optimization problems in order to provide users with the most efficient ways to exploit available resources. In this work we present an overview of our current research activities concerning the development of new algorithms, based on CLP techniques, for route planning problems exploiting the geometric information intrinsically present in many of them or in some of their variants. The research so far has focused in particular on the Euclidean Traveling Salesperson Problem (Euclidean TSP) with the aim to exploit the results obtained also to other problems of the same category, such as the Euclidean Vehicle Routing Problem (Euclidean VRP), in the future.Comment: In Proceedings ICLP 2020, arXiv:2009.0915

TSP – Infrastructure for the Traveling Salesperson Problem

The traveling salesperson (or, salesman) problem (TSP) is a well known and important combinatorial optimization problem. The goal is to find the shortest tour that visits each city in a given list exactly once and then returns to the starting city. Despite this simple problem statement, solving the TSP is difficult since it belongs to the class of NP-complete problems. The importance of the TSP arises besides from its theoretical appeal from the variety of its applications. Typical applications in operations research include vehicle routing, computer wiring, cutting wallpaper and job sequencing. The main application in statistics is combinatorial data analysis, e.g., reordering rows and columns of data matrices or identifying clusters. In this paper we introduce the R˜package TSP which provides a basic infrastructure for handling and solving the traveling salesperson problem. The package features S3 classes for specifying a TSP and its (possibly optimal) solution as well as several heuristics to find good solutions. In addition, it provides an interface to Concorde, one of the best exact TSP solvers currently available. Keywords:˜combinatorial optimization, traveling salesman problem, R. 1