Improving Christofides' lower bound for the traveling salesman problem

In 1972 Christofides introduced a lower bound for the Traveling Salesman Problem (TSP). The bound is based on solving repeatedly a Linear Assignment Problem. We relate the bound to the Complete Cycle Problem; as a consequence the correctness of the bound is easier to prove.

Further we give improvements for the bound in the symmetric case and we deal with the influence of the triangle equation together with the identification of non-optimal edges for the TSP. The improvements are illustrated by examples and computational results for large problems.     

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.     

The on-line asymmetric traveling salesman problem

We consider two on-line versions of the asymmetric traveling salesman problem with triangle inequality. For the homing version, in which the salesman is required to return in the city where it started from, we give a $\frac{3+\sqrt{5}}{2}$-competitive algorithm and prove that this is best possible. For the nomadic version, the on-line analogue of the shortest asymmetric Hamiltonian path problem, we show that the competitive ratio of any on-line algorithm depends on the amount of asymmetry of the space in which the salesman moves. We also give bounds on the competitive ratio of on-line algorithms that are zealous, that is, in which the salesman cannot stay idle when some city can be served.     

Cluster based branching for the asymmetric traveling salesman problem

This paper presents a new branching scheme for the asymmetric traveling salesman problem (ATSP) based on clusters. A cluster is defined as a node set with the characteristic that there exists an optimal solution in which the nodes in the node set are visited consecutively. The paper considers identification of clusters, implementation of a cluster based branching scheme, and cluster based dominance tests. The new approach is implemented in a branch and bound algorithm using a well-known additive bounding procedure. Considerable savings in computing time are obtained compared to previously published assignment based branch and bound algorithms for the ATSP.     

A new lower bound for the geometric traveling salesman problem in terms of discrepancy

We give a new lower bound for the shortest hamiltonian path through n points of [0,1]^d in terms of the discrepancy of these n points. This improves an earlier result by Steele.     

Iterative patching and the asymmetric traveling salesman problem

Although Branch-and-Bound (BnB) methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. In this paper, we investigate iterative patching, a technique in which a fixed patching procedure is applied at each node of the BnB search tree for the Asymmetric Traveling Salesman Problem. Computational experiments show that iterative patching results in general in search trees that are smaller than the classical BnB trees, and that solution times are lower for usual random and sparse instances. Furthermore, it turns out that, on average, iterative patching with the Contract-or-Patch procedure of Glover, Gutin, Yeo and Zverovich (2001) and the Karp–Steele procedure are the fastest, and that ‘iterative’ Modified Karp–Steele patching generates the smallest search trees.     

A simple LP relaxation for the asymmetric traveling salesman problem

It is a long-standing open question in combinatorial optimization whether the integrality gap of the subtour linear program relaxation (subtour LP) for the asymmetric traveling salesman problem (ATSP) is a constant. The study on the structure of this linear program is important and extensive. In this paper, we give a new and simpler LP relaxation for the ATSP. Our linear program consists of a single type of constraints that combine both the subtour elimination and the degree constraints in the traditional subtour LP. As a result, we obtain a much simpler relaxation. In particular, it is shown that the extreme solutions of our program have at most 2n ? 2 non-zero variables, improving the bound 3n ? 2, proved by Vempala and Yannakakis, for the ones obtained by the subtour LP. Nevertheless, the integrality gap of the new linear program is larger than that of the traditional subtour LP by at most a constant factor.     

Genetic algorithm for asymmetric traveling salesman problem with imprecise travel times

This paper presents a variant of the asymmetric traveling salesman problem (ATSP) in which the traveling time between each pair of cities is represented by an interval of values (wherein the actual travel time is expected to lie) instead of a fixed (deterministic) value as in the classical ATSP. Here the ATSP (with interval objective) is formulated using the usual interval arithmetic. To solve the interval ATSP (I-ATSP), a genetic algorithm with interval valued fitness function is proposed. For this purpose, the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used. The proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators. To analyze the performance and effectiveness of the proposed algorithm and different genetic operators, computational studies of the proposed algorithm on some randomly generated test problems are reported.     

Application of the out-of-kilter algorithm to the asymmetric traveling salesman problem

This paper presents a heuristic method that finds optimum or near-optimum solutions to the asymmetric traveling salesman problem. The method uses the out-of-kilter algorithm to search for a neighbourhood. When subtours are produced by a flow-augmenting path of the out-of-kilter algorithm, it patches them into a Hamiltonian cycle. It extends the neighbourhood space by exchanging an even number of arcs, and it also exchanges arcs by a non-sequential primary change. Instances from real applications were used to test the algorithm, along with randomly generated problems. The new heuristic algorithm produced optimum solutions for 16 out of 28 real-world instances from TSPLIB and other sources. Also, compared with four efficient heuristics, it produced the best solutions for all except six instances. It also produced relatively good solutions in reasonable times for 216 randomly generated instances from nine instance generators.     

A parallel branch and bound algorithm for solving large asymmetric traveling salesman problems

A parallel branch and bound algorithm that solves the asymmetric traveling salesman problem to optimality is described. The algorithm uses an assignment problem based lower bounding technique, subtour elimination branching rules, and a subtour patching algorithm as an upper bounding procedure. The algorithm is organized around a data flow framework for parallel branch and bound. The algorithm begins by converting the cost matrix to a sparser version in such a fashion as to retain the optimality of the final solution. Computational results are presented for three different classes of problem instances: (1) matrix elements drawn from a uniform distribution of integers for instances of size 250 to 10 000 cities, (2) instances of size 250 to 1000 cities that concentrate small elements in the upper left portion of the cost matrix, and (3) instances of size 300 to 3000 cities that are designed to confound neighborhood search heuristics.     

A new asymmetric pyramidally solvable class of the traveling salesman problem

We present a new method of identifying a class of asymmetric matrices for which an optimal traveling salesman tour exists that is pyramidal. The new class generalizes two previously known classes of matrices and includes some new matrices as well.     

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].     

New formulation for the high multiplicity asymmetric traveling salesman problem with application to the Chesapeake problem

This paper presents a new model for a special type of traveling salesman problem called the High Multiplicity Asymmetric Traveling Salesman Problem (HMATSP). The formulation adopts a flow-based subtour elimination structure and establishes its validity for this problem. Also, we present computational results to demonstrate the efficacy of our modeling approach. The model is then incorporated as a substructure in a formulation for the lot-sizing problem involving parallel machines and sequence-dependent setup costs, also known as the Chesapeake Problem, and related test problems are solved to optimality for the first time in the literature.     

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.     

The traveling salesman problem: a book review

We review the recent book authored by David L. Applegate, Robert E. Bixby, Vasěk Chvátal and William J. Cook, The traveling salesman problem: a computational study, Princeton Series in Applied Mathematics. Princeton University Press 2007, Hardback price $45.00 / ￡26.95, 606pp, ISBN 978-0-691-12993-8.      

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.     

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.     

Diversified local search strategy under scatter search framework for the probabilistic traveling salesman problem

This paper focuses on introducing a concept of diversified local search strategy under the scatter search framework for the probabilistic traveling salesman problem (PTSP). Different combinations of three commonly used local search methods in the PTSP, i.e., 1-shift, 2-opt, and 3-opt, were used to investigate its effects. A set of numerical experiments were conducted to test the validity of the proposed strategy based on randomly generated test instances. The numerical results and the permutation test showed that the diversified local search strategy, especially by combining 1-shift and 2-opt algorithms, can most effectively solve the homogeneous and heterogeneous PTSP in most of the tested instances in comparison with the single local search strategy under scatter search framework.