Traveling salesman games

In this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited. We introduce two types of cooperative games—fixed-route traveling salesman games and traveling salesman games—as a tool to attack this problem. Under very mild conditions we prove that fixed-route traveling salesman games have non-empty cores if the fixed route is a solution of the classical traveling salesman problem. Core elements provide us with fair cost allocations. A traveling salesman game may have an empty core, even if the cost matrix satisfies the triangle inequality. In this paper we introduce a class of matrices defining TS-games with non-empty cores.     

Submodularity and the traveling salesman problem

In this paper we investigate the relationship between traveling salesman tour lengths and submodular functions. This work is motivated by the one warehouse multi-retailer inventory/distribution problem with traveling salesman tour vehicle routing costs. Our goal is to find a submodular function whose values are close to those of optimal tour lengths through a central warehouse and a group of retailers. Our work shows that a submodular approximation to traveling salesman tour lengths whose error is bounded by a constant does not exist. However, we present heuristics that have errors which grow slowly with the number of retailers for the traveling salesman problem in the Euclidean plane. Furthermore, we perform computational tests that show that our submodular approximations of traveling salesman tour lengths have smaller errors than our theoretical worst case analysis would lead us to believe.     

The adjacency relation on the traveling salesman polytope is NP-Complete

We consider the problem of determining whether two traveling salesman tours correspond to non-adjacent vertices of the convex polytope associated with the traveling salesman problem. This problem is shown to be NP-Complete for both the symmetric and nonsymmetric traveling salesman problem. Several implications are discussed.This Research was supported by NSF Grant GK-420488, the U.S. Army Research Office-Durham under Grant DAHC04-75-G0192, and an IBM Fellowship.     

Expanding neighborhood search–GRASP for the probabilistic traveling salesman problem

The Probabilistic Traveling Salesman Problem is a variation of the classic traveling salesman problem and one of the most significant stochastic routing problems. In probabilistic traveling salesman problem only a subset of potential customers need to be visited on any given instance of the problem. The number of customers to be visited each time is a random variable. In this paper, a variant of the well-known Greedy Randomized Adaptive Search Procedure (GRASP), the Expanding Neighborhood Search–GRASP, is proposed for the solution of the probabilistic traveling salesman problem. expanding neighborhood search–GRASP has been proved to be a very efficient algorithm for the solution of the traveling salesman problem. The proposed algorithm is tested on a numerous benchmark problems from TSPLIB with very satisfactory results. Comparisons with the classic GRASP algorithm and with a Tabu Search algorithm are also presented. Also, a comparison is performed with the results of a number of implementations of the Ant Colony Optimization algorithm from the literature and in six out of ten cases the proposed algorithm gives a new best solution.     

Solving a multiobjective traveling salesman problem by dynamic programming

In this paper Klötzler's method of multiobjective dynamic programming is applied to the solution of a two-dimensional traveling salesman problem. In this way Bellman's and Held/Karp's dynamic programming approach to one-dimensional traveling salesman problems is extended to the multiobjective case.     

Quality inspection scheduling for multi-unit service enterprises

The traveling salesman problem is a classic NP-hard problem used to model many production and scheduling problems. The problem becomes even more difficult when additional salesmen are added to create a multiple traveling salesman problem (MTSP). We consider a variation of this problem where one salesman visits a given set of cities in a series of short trips. This variation is faced by numerous franchise companies that use quality control inspectors to ensure properties are maintaining acceptable facility and service levels. We model an actual franchised hotel chain using traveling quality inspectors to demonstrate the technique. The model is solved using a commercially available genetic algorithm (GA) tool as well as a custom GA program. The custom GA is proven to be an effective method of solving the proposed model.     

求解旅行商问题的一种改进粒子群算法

本文研究了求解旅行商问题的粒子群算法。针对标准粒子群算法在求解旅行商问题过程中容易出现早熟和停滞现象的缺点,提出了一种改进的粒子群算法。首先,在初始种群的选取过程中,利用改进的贪婪策略直接获得具有较高性能的初始种群以提高算法的搜索效率。其次,通过引入次优吸引子,使粒子在搜索过程中可以更加充分地利用群体的信息来提高自身的性能,有效抑制收敛过程中的停滞现象,提高算法的搜索能力。最后为了验证所提出的方法的有效性和可行性,对TSPLIB标准库中的多个实例进行了测试,并给出了数值结果。     

Genetic algorithms for the traveling salesman problem

This paper is a survey of genetic algorithms for the traveling salesman problem. Genetic algorithms are randomized search techniques that simulate some of the processes observed in natural evolution. In this paper, a simple genetic algorithm is introduced, and various extensions are presented to solve the traveling salesman problem. Computational results are also reported for both random and classical problems taken from the operations research literature.     

An improved assignment lower bound for the euclidean traveling salesman problem

A simple transformation of the distance matrix for the Euclidean traveling salesman problem is presented that produces a tighter lower bound on the length of the optimal tour than has previously been attainable using the assignment relaxation. The improved lower bound is obtained by exploiting geometric properties of the problem to produce fewer and larger subtours on the first solution of the assignment problem. This research should improve the performance of assignment based exact procedures and may lead to improved heuristics for the traveling salesman problem.     

A comparison of lower bounds for the symmetric circulant traveling salesman problem

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare different lower bounds on the optimal value that may be computed in polynomial time. We derive a new linear programming (LP) relaxation of the SCTSP from the semidefinite programming (SDP) relaxation in [E. de Klerk, D.V. Pasechnik, R. Sotirov, On semidefinite programming relaxation of the traveling salesman problem, SIAM Journal of Optimization 19 (4) (2008) 1559-1573]. Further, we discuss theoretical and empirical comparisons between this new bound and three well-known bounds from the literature, namely the Held-Karp bound [M. Held, R.M. Karp, The traveling salesman problem and minimum spanning trees, Operations Research 18 (1970) 1138-1162], the 1-tree bound, and the closed-form bound for SCTSP proposed in [J.A.A. van der Veen, Solvable cases of TSP with various objective functions, Ph.D. Thesis, Groningen University, The Netherlands, 1992].     

A new extension principle algorithm for the traveling salesman problem

In this paper, an algorithm for solving the asymmetric traveling salesman problem is developed and tested by computation. This algorithm is based on the extension principle by Schoch and uses the assignment problem relaxation of the traveling salesman problem for computing lower bounds. Computational experience with randomly generated test problems indicate that the present algorithm yields good results in runtime which are comparable with the results of Smith/Srinivasan/Thompson. Computational experience are reported for up to 120-node problems with uniformly distributed and approximately normally distributed cost.     

The symmetric quadratic traveling salesman problem

In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e.g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmetric case, costs do not depend on the direction of traversal. We study the polyhedral structure of a linearized integer programming formulation of the symmetric quadratic traveling salesman problem. Our constructive approach for establishing the dimension of the underlying polyhedron is rather involved but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the Boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic strengthening approach for lifting valid inequalities of the usual traveling salesman problem to stronger valid inequalities for the symmetric quadratic traveling salesman problem. Applying this strengthening to subtour elimination constraints gives rise to facet defining inequalities, but finding a maximally violated inequality among these is NP-complete. For the simplest comb inequality with three teeth the strengthening is no longer sufficient to obtain a facet. Preliminary computational results indicate that the new cutting planes may help to considerably improve the quality of the root relaxation in some important applications.     

Continuous line drawings via the traveling salesman problem

We describe how to use the traveling salesman problem to create continuous line drawings of target pictures.     

The symmetric traveling salesman problem and edge exchanges in minimal 1-trees

In this article the effect of exchanging edges inside a minimal 1-tree with edges outside is analysed. In combination with an upper bound this analysis enables the elimination of variables in the symmetric traveling salesman problem. After discussion on a number of improvements for this analysis, the implementation is described in a traveling salesman algorithm based on the 1-tree relaxation. Computational results show the advantages of the edges exchanges for Euclidean problems (up to 120 cities) as well as for random table problems (up to 200 cities).     

The delivery man problem on a tree network

This note considers a variant of the traveling salesman problem in which we seek a route that minimizes the total of the vertex arrival times. This problem is called the deliverly man problem. The traveling salesman problem is NP-complete on a general network and trivial on a tree network. The delivery man problem is also NP-complete on a general network but far from trivial on a tree network. Various characteristics of the delivery man problem for tree networks are explored and a pseudo-polynomial time solution algorithm is proposed.This research was sponsored by NSF Grant #ECS-8104647.     

A tabu search heuristic using genetic diversification for the clustered traveling salesman problem

The clustered traveling salesman problem is an extension of the classical traveling salesman problem where the set of vertices is partitioned into clusters. The objective is to find a least cost Hamiltonian cycle such that the vertices of each cluster are visited contiguously and the clusters are visited in a prespecified order. A tabu search heuristic is proposed to solve this problem. This algorithm periodically restarts its search by merging two elite solutions to form a new starting solution (in a manner reminiscent of genetic algorithms). Computational results are reported on sets of Euclidean problems with different characteristics.     

NP-completeness of the hamming salesman problem

It is shown that the traveling salesman problem, where cities are bit strings with Hamming distances, is NP-complete.     

Efficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problem

The generalized traveling salesman problem (GTSP) is a well-known combinatorial optimization problem with a host of applications. It is an extension of the Traveling Salesman Problem (TSP) where the set of cities is partitioned into so-called clusters, and the salesman has to visit every cluster exactly once.