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 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.     

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.     

Four-point conditions for the TSP: The complete complexity classification

The combinatorial optimization literature contains a multitude of polynomially solvable special cases of the traveling salesman problem (TSP) which result from imposing certain combinatorial restrictions on the underlying distance matrices. Many of these special cases have the form of so-called four-point conditions: inequalities that involve the distances between four arbitrary cities.In this paper we classify all possible four-point conditions for the TSP with respect to computational complexity, and we determine for each of them whether the resulting special case of the TSP can be solved in polynomial time or whether it remains NP-hard.     

The open shop problem with routing at a two-node network and allowed preemption

The open shop problem with routing and allowed preemption is a generalization of the two classical discrete optimization problems: the NP-hard metrical traveling salesman problem and the polynomially solvable scheduling problem, i.e., the open shop with allowed preemption. In the paper, a partial case of this problem is considered when the transportation network consists of two nodes. It is proved that the problem with two machines is polynomially solvable, while the problem is NP-hard in the strong sense in the case of not fixed number of machines.     

A random-key genetic algorithm for the generalized traveling salesman problem

The generalized traveling salesman problem is a variation of the well-known traveling salesman problem in which the set of nodes is divided into clusters; the objective is to find a minimum-cost tour passing through one node from each cluster. We present an effective heuristic for this problem. The method combines a genetic algorithm (GA) with a local tour improvement heuristic. Solutions are encoded using random keys, which circumvent the feasibility problems encountered when using traditional GA encodings. On a set of 41 standard test problems with symmetric distances and up to 442 nodes, the heuristic found solutions that were optimal in most cases and were within 1% of optimality in all but the largest problems, with computation times generally within 10 seconds. The heuristic is competitive with other heuristics published to date in both solution quality and computation time.     

Local search for the probabilistic traveling salesman problem: Correction to the 2-p-opt and 1-shift algorithms

The probabilistic traveling salesman problem concerns the best way to visit a set of customers located in some metric space, where each customer requires a visit only with some known probability. A solution to this problem is an a priori tour which visits all customers, and the objective is to minimize the expected length of the a priori tour over all customer subsets, assuming that customers in any given subset must be visited in the same order as they appear in the a priori tour. This problem belongs to the class of stochastic vehicle routing problems, a class which has received increasing attention in recent years, and which is of major importance in real world applications.Several heuristics have been proposed and tested for the probabilistic traveling salesman problem, many of which are a straightforward adaptation of heuristics for the classical traveling salesman problem. In particular, two local search algorithms (2-p-opt and 1-shift) were introduced by Bertsimas.In a previous report we have shown that the expressions for the cost evaluation of 2-p-opt and 1-shift moves, as proposed by Bertsimas, are not correct. In this paper we derive the correct versions of these expressions, and we show that the local search algorithms based on these expressions perform significantly better than those exploiting the incorrect expressions.     

Match twice and stitch: a new TSP tour construction heuristic

We present a new symmetric traveling salesman problem tour construction heuristic. Two sequential matchings yield a set of cycles over the given point set; these are then stitched to form a tour. Our method outperforms all previous tour construction methods, but is dominated by several tour improvement heuristics.     

Multiobjective traveling salesperson problem on Halin graphs

In this paper, we study traveling salesperson (TSP) and bottleneck traveling salesperson (BTSP) problems on special graphs called Halin graphs. Although both problems are NP-Hard on general graphs, they are polynomially solvable on Halin graphs. We address the multiobjective versions of these problems. We show computational complexities of finding a single nondominated point as well as finding all nondominated points for different objective function combinations. We develop algorithms for the polynomially solvable combinations.     

On inverse traveling salesman problems

The inverse traveling salesman problem belongs to the class of ??inverse combinatorial optimization?? problems. In an inverse combinatorial optimization problem, we are given a feasible solution for an instance of a particular combinatorial optimization problem, and the task is to adjust the instance parameters as little as possible so that the given solution becomes optimal in the new instance. In this paper, we consider a variant of the inverse traveling salesman problem, denoted by ITSP _W,A , by taking into account a set W of admissible weight systems and a specific algorithm. We are given an edge-weighted complete graph (an instance of TSP), a Hamiltonian tour (a feasible solution of TSP) and a specific algorithm solving TSP. Then, ITSP _W,A , is the problem to find a new weight system in W which minimizes the difference from the original weight system so that the given tour can be selected by the algorithm as a solution. We consider the cases ${W \in \{\mathbb{R}^{+m}, \{1, 2\}^m , \Delta\}}$ where ?? denotes the set of edge weight systems satisfying the triangular inequality and m is the number of edges. As for algorithms, we consider a local search algorithm 2-opt, a greedy algorithm closest neighbor and any optimal algorithm. We devise both complexity and approximation results. We also deal with the inverse traveling salesman problem on a line for which we modify the positions of vertices instead of edge weights. We handle the cases ${W \in \{\mathbb{R}^{+n}, \mathbb{N}^n\}}$ where n is the number of vertices.     

Algorithms for the universal and a priori TSP

We present two simple results for generalizations of the traveling salesman problem (TSP): for the universal TSP, we show that one can compute a tour that is universally optimal whenever the input is a tree metric. A (randomized) O(logn)-approximation algorithm for the a priori TSP follows as a corollary.     

Solvable Cases of the No-Wait Flow-Shop Scheduling Problem

The no-wait flow-shop scheduling problem (NWFSSP) with a makespan objective function is considered. As is well known, this problem is ????-hard for three or more machines. Therefore, it is interesting to consider special cases, i.e. special structured processing time matrices, that allow polynomial time solution algorithms. Furthermore, it is well known that the NWFSSP with a makespan objective function can be formulated as a travelling salesman problem (TSP). It is observed that special structured processing time matrices for the NWFSSP lead to special structured distance matrices for which the TSP is polynomially solvable. Using this observation, it is shown that some NWFSSPs with fixed processing times on all except two machines are well solvable while the others are conjectured to be ????-hard. Also, it is shown that NWFSSPs with a mean completion time objective function restricted to semi-ordered processing time matrices are easily solvable.     

Heuristics for unequal weight delivery problems with a fixed error guarantee

This paper presents heuristics that are based on optimal partitioning of a travelling salesman tour, for solving the unequal weight delivery problem. The worst case error performance is given as a bound on the worst case ratio of the cost of the heuristic solution to the cost of the optimal solution. A fully polynomial procedure which consists of applying the optimal partitioning to a travelling salesman tour generated by Christofides' heuristic has a worst case error bound of 3.5−3/Q where Q is the capacity limit of the vehicles. When optimal partitioning is applied to an optimal travelling salesman tour, the worst case error bound becomes 3−2/Q.     

On the solutions of stochastic traveling salesman problems

We consider the n-city traveling salesman problem where the distances between the cities are nondeterministic. Our purpose is to estimate the expectation of the length of the optimal tour. This is done by calculating the expectations of a lower bound and an upper bound for the length of the optimal tour. Because the upper bound is formed by the well-known nearest neighbour rule, we can simultaneously find the cases where this rule is effective in the mean. If we let the number of cities grow, we obtain symptotic results that are totally determined by the behaviour of the distribution of the distance between any two points in the neighbourhood of the distance zero.     

Worst-case analysis of two travelling salesman heuristics

In this paper we analyze the worst-case performance of some heuristics for the symmetric travelling salesman problem. We show that the worst-case ratios of tour length produced by the savings and greedy heuristics to that of a minimum tour are bounded by [log₂n]+1 and 0.5([log₂n]+1) respectively, where n is the number of cities.     

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.     

On estimating the distribution of optimal traveling salesman tour lengths using heuristics

The traveling salesman problem is an important combinatorial optimization problem due to its significance in academic research and its real world applications. The problem has been extensively studied and much is known about its polyhedral structure and algorithms for exact and heuristic solutions. While most work is concentrated on solving the deterministic version of the problem, there also has been some research on the stochastic TSP. Research on the stochastic TSP has concentrated on asymptotic properties and estimation of the TSP-constant. Not much is, however, known about the probability distribution of the optimal tour length. In this paper, we present some empirical results based on Monte Carlo simulations for the symmetric Euclidean and Rectilinear TSPs. We derive regression equations for predicting the first four moments of the distribution of estimated TSP tour lengths using heuristics. We then show that a Beta distribution gives excellent fits for small to moderate sized TSP problems. We derive regression equations for predicting the parameters of the Beta distribution. Finally we predict the TSP constant using two alternative approaches.     

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.     

On no-wait and no-idle flow shops with makespan criterion

The paper deals with the NP-hard problems of minimizing the makespan in m-machine no-wait and no-idle permutation flow shops. We identify networks whose longest path lengths represent the makespans. These networks reveal the duality between the two problems, and show graphical explanations of the fact that under no-wait and no-idle conditions the makespan can be a decreasing function of some job processing times. Moreover, they also lead to a natural reduction of the no-wait flow shop problem to the traveling salesman problem, some lower bounds on the shortest makespan, and new efficiently solvable special cases.     

The pyramidal capacitated vehicle routing problem

This paper introduces the pyramidal capacitated vehicle routing problem (PCVRP) as a restricted version of the capacitated vehicle routing problem (CVRP). In the PCVRP each route is required to be pyramidal in a sense generalized from the pyramidal traveling salesman problem (PTSP). A pyramidal route is defined as a route on which the vehicle first visits customers in increasing order of customer index, and on the remaining part of the route visits customers in decreasing order of customer index.