POPMUSIC for the point feature label placement problem

Point feature label placement is the problem of placing text labels adjacent to point features on a map so as to maximize legibility. The goal is to choose positions for the labels that do not give rise to label overlaps and that minimize obscuration of features. A practical goal is to minimize the number of overlaps while considering cartographic preferences. This article proposes a new heuristic for solving the point feature label placement problem based on the application of the POPMUSIC frame. Computational experiments show that the proposed heuristic outperformed other recent metaheuristics approaches in the literature. Experiments with problem instances involving up to 10 million points show that the computational time of the proposed heuristic increases almost linearly with the problem size. New problem instances based on real data with more than 13,000 labels are proposed.     

A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem

We present a variable neighborhood search approach for solving the one-commodity pickup-and-delivery travelling salesman problem. It is characterized by a set of customers such that each of the customers either supplies (pickup customers) or demands (delivery customers) a given amount of a single product, and by a vehicle, whose given capacity must not be exceeded, that starts at the depot and must visit each customer only once. The objective is to minimize the total length of the tour. Thus, the considered problem includes checking the existence of a feasible travelling salesman’s tour and designing the optimal travelling salesman’s tour, which are both NP-hard problems. We adapt a collection of neighborhood structures, k-opt, double-bridge and insertion operators mainly used for solving the classical travelling salesman problem. A binary indexed tree data structure is used, which enables efficient feasibility checking and updating of solutions in these neighborhoods. Our extensive computational analysis shows that the proposed variable neighborhood search based heuristics outperforms the best-known algorithms in terms of both the solution quality and computational efforts. Moreover, we improve the best-known solutions of all benchmark instances from the literature (with 200 to 500 customers). We are also able to solve instances with up to 1000 customers.     

A bilevel programming approach to the travelling salesman problem

We show that the travelling salesman problem is polynomially reducible to a bilevel toll optimization program. Based on natural bilevel programming techniques, we recover the lifted Miller-Tucker-Zemlin constraints. Next, we derive an O(n²) multi-commodity extension whose LP relaxation is comparable to the exponential formulation of Dantzig, Fulkerson and Johnson.     

A new class of cutting planes for the symmetric travelling salesman problem

A comprehensive class of cutting planes for the symmetric travelling salesman problem (TSP) is proposed which contains the known comb inequalities, the path inequalities and the 3-star constraints as special cases. Its relation to the clique tree inequalities is discussed. The cutting planes are shown to be valid for a relaxed version of the TSP, the travelling salesman problem on a road network, and—under certain conditions—to define facets of the polyhedron associated with this problem.     

Facets of the polytope of the asymmetric travelling salesman problem with replenishment arcs

The Asymmetric Travelling Salesman Problem with Replenishment Arcs (RATSP) is a new class of problems arising from work related to aircraft routing. Given a digraph with cost on the arcs, a solution of the RATSP, like that of the Asymmetric Travelling Salesman Problem, induces a directed tour in the graph which minimises total cost. However the tour must satisfy additional constraints: the arc set is partitioned into replenishment arcs and ordinary arcs, each node has a non-negative weight associated with it, and the tour cannot accumulate more than some weight limit before a replenishment arc must be used. To enforce this requirement, constraints are needed. We refer to these as replenishment constraints.In this paper, we review previous polyhedral results for the RATSP and related problems, then prove that two classes of constraints developed in V. Mak and N. Boland [Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs, Technical Report TR M05/03, School of Information Technology, Deakin University, 2005] are, under appropriate conditions, facet-defining for the RATS polytope.     

The biobjective travelling purchaser problem

The purpose of this article is to present and solve the Biobjective Travelling Purchaser Problem, which consists in determining a route through a subset of markets in order to collect a set of products, minimizing the travel distance and the purchasing cost simultaneously. The most convenient purchase of the product in the visited markets is easily computed once the route has been determined. Therefore, this problem contains a finite set of solutions (one for each route) and the problem belongs to the field of the Biobjective Combinatorial Optimization. It is here formulated as a Biobjective Mixed Integer Linear Programming model with an exponential number of valid inequalities, and this model is used within a cutting plane algorithm to generate the set of all supported and non-supported efficient points in the objective space. A variant of the algorithm computes only supported efficient points. For each efficient point in the objective space exactly one Pareto optimal solution in the decision space is computed by solving a single-objective problem. Each of these single-objective problems, in turn, is solved by a specific branch-and-cut approach. A heuristic improvement based on saving previously generated cuts in a common cut-pool structure has also been developed with the aim of speeding up the algorithm performance. Results based on benchmark instances from literature show that the common cut-pool heuristic is very useful, and that the proposed algorithm manages to solve instances containing up to 100 markets and 200 different products. The general procedure can be extended to address other biobjective combinatorial optimization problems whenever a branch-and-cut algorithm is available to solve a single-objective linear combination of these criteria.     

On patching algorithms for random asymmetric travelling salesman problems

Let the arc-lengthsL _ij of a complete digraph onn vertices be independent uniform [0, 1] random variables. We consider the patching algorithm of Karp and Steele for the travelling salesman problem on such a digraph and give modifications which tighten the expected error. We extend these ideas to thek-person travelling salesman problem and also consider the case where cities can be visited more than once.     

On the empirical scaling of run-time for finding optimal solutions to the travelling salesman problem

The travelling salesman problem (TSP)   is one of the most prominent NP-hard combinatorial optimisation problems. After over fifty years of intense study, the TSP continues to be of broad theoretical and practical interest. Using a novel approach to empirical scaling analysis, which in principle is applicable to solvers for many other problems, we demonstrate that some of the most widely studied types of TSP instances tend to be much easier than expected from previous theoretical and empirical results. In particular, we show that the empirical median run-time required for finding optimal solutions to so-called random uniform Euclidean (RUE) instances – one of the most widely studied classes of TSP instances – scales substantially better than Θ(2ⁿ)$\Theta (2^n)$ with the number n of cities to be visited. The Concorde solver, for which we achieved this result, is the best-performing exact TSP solver we are aware of, and has been applied to a broad range of real-world problems. Furthermore, we show that even when applied to a broad range of instances from the prominent TSPLIB benchmark collection for the TSP, Concorde exhibits run-times that are surprisingly consistent with our empirical model of Concorde’s scaling behaviour on RUE instances. This result suggests that the behaviour observed for the simple random structure underlying RUE is very similar to that obtained on the structured instances arising in various applications.     

Maximum travelling salesman problem

We present different types of techniques for designing algorithms with worst-case performances for the Maximum Travelling Salesman Problem. Supported by Byelarussian Fundamental Science Found and DAAD     

Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs

The asymmetric travelling salesman problem with replenishment arcs (RATSP), arising from work related to aircraft routing, is a generalisation of the well-known ATSP. In this paper, we introduce a polynomial size mixed-integer linear programming (MILP) formulation for the RATSP, and improve an existing exponential size ILP formulation of Zhu [The aircraft rotation problem, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, 1994] by proposing two classes of stronger cuts. We present results that under certain conditions, these two classes of stronger cuts are facet-defining for the RATS polytope, and that ATSP facets can be lifted, to give RATSP facets. We implement our polyhedral findings and develop a Lagrangean relaxation (LR)-based branch-and-bound (BNB) algorithm for the RATSP, and compare this method with solving the polynomial size formulation using ILOG Cplex 9.0, using both randomly generated problems and aircraft routing problems. Finally we compare our methods with the existing method of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408-427]. It turns out that both of our methods are much faster than that of Boland et al. [The asymmetric traveling salesman problem with replenishment arcs, European J. Oper. Res. 123 (2000) 408-427], and that the LR-based BNB method is more efficient for problems that resemble the aircraft rotation problems.     

Selective generalized travelling salesman problem

ABSTRACT

This paper introduces the Selective Generalized Traveling Salesman Problem (SGTSP). In SGTSP, the goal is to determine the maximum profitable tour within the given threshold of the tour’s duration, which consists of a subset of clusters and a subset of nodes in each cluster visited on the tour. This problem is a combination of cluster and node selection and determining the shortest path between the selected nodes. We propose eight mixed integer programming (MIP) formulations for SGTSP. All of the given MIP formulations are completely new, which is one of the major novelties of the study. The performance of the proposed formulations is evaluated on a set of test instances by conducting 4608 experimental runs. Overall, 4138 out of 4608 (~90%) test instances were solved optimally by using all formulations.     

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.     

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.     

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.     

Risky traveling salesman problem

In this paper we introduce a methodology for optimizing the expected cost of routing a single vehicle which has a probability of breaking down or failing to complete some of its tasks. More specifically, a calculus is devised for finding the optimal order in which each site should be visited.     

Guaranteed performance heuristics for the bottleneck travelling salesman problem

We consider constant-performance, polynomial-time, nonexact algorithms for the minimax or bottleneck version of the Travelling Salesman Problem. It is first shown that no such algorithm can exist for problems with arbitrary costs unless P = NP. However, when costs are positive and satisfy the triangle inequality, we use results pertaining to the squares of biconnected graphs to produce a polynomial-time algorithm with worst-case bound 2 and show further that, unless P = NP, no polynomial alternative can improve on this value.     

Approximate algorithms for the travelling purchaser problem

In this paper we study the generalized savings heuristics of Golden, Levy and Dahl and propose several new heuristic procedures for solving the travelling purchaser problem. A comparative study of the four heuristics considered is provided.     

The x-and-y-axes travelling salesman problem

The x-and-y-axes travelling salesman problem forms a special case of the Euclidean TSP, where all cities are situated on the x-axis and on the y-axis of an orthogonal coordinate system of the Euclidean plane. By carefully analyzing the underlying combinatorial and geometric structures, we show that this problem can be solved in polynomial time. The running time of the resulting algorithm is quadratic in the number of cities.