Stability aspects of the traveling salesman problem based on k-best solutions

AbstractThis paper discusses stability analysis for the Traveling Salesman Problem (TSP). For a traveling salesman tour which is known to be optimal with respect to a given instance (length vector) we are interested in determining the stability region, i.e. the set of all length vectors for which the tour is optimal. The following three subsets of the stability region are of special interest: 1.(1) tolerances, i.e. the maximum perturbations of single edges;2.(2) tolerance regions which are subsets of the stability region that can be constructed from the tolerances; and3.(3) the largest ball contained in the stability region centered at the given length vector (the corresponding radius is known as the stability radius). It is well known that the problems of determining tolerances and the stability radius for the TSP are NP-hard so that in general it is not possible to obtain the above-mentioned three subsets without spending a lot of computation time. The question addressed in this paper is the following: assume that not only an optimal tour is known, but also a set of k shortest tours (k ⩾2) is given. Then to which extent does this allow us to determine the three subsets in polynomial time? It will be shown in this paper that having k-best solutions can give the desired information only partially. More precisely, it will be shown that only some of the tolerances can be determined exactly and for the other ones as well as for the stability radius only lower and/or upper bounds can be derived. Since the amount of information that can be derived from the set of k-best solutions is dependent on both the value of k as well as on the specific length vector, we present numerical experiments on instances from the TSPLIB library to analyze the effectiveness of our approach

A Hybrid Genetic Algorithm for the Traveling Salesman Problem with Drone

This paper addresses the Traveling Salesman Problem with Drone (TSP-D), in which a truck and drone are used to deliver parcels to customers. The objective of this problem is to either minimize the total operational cost (min-cost TSP-D) or minimize the completion time for the truck and drone (min-time TSP-D). This problem has gained a lot of attention in the last few years since it is matched with the recent trends in a new delivery method among logistics companies. To solve the TSP-D, we propose a hybrid genetic search with dynamic population management and adaptive diversity control based on a split algorithm, problem-tailored crossover and local search operators, a new restore method to advance the convergence and an adaptive penalization mechanism to dynamically balance the search between feasible/infeasible solutions. The computational results show that the proposed algorithm outperforms existing methods in terms of solution quality and improves best known solutions found in the literature. Moreover, various analyses on the impacts of crossover choice and heuristic components have been conducted to analysis further their sensitivity to the performance of our method.Comment: Technical Report. 34 pages, 5 figure

Playing Billiard in Version Space

A ray-tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a set of linear separable examples. While the Bayes-optimum requires a majority decision over all Perceptrons separating the example set, the problem considered here corresponds to finding the single Perceptron with best average generalization probability. For randomly distributed examples the billiard estimate agrees with known analytic results. In real-life classification problems the generalization error is consistently reduced compared to the maximal stability Perceptron.Comment: uuencoded, gzipped PostScript file, 127576 bytes To recover 1) save file as bayes.uue. Then 2) uudecode bayes.uue and 3) gunzip bayes.ps.g

The AddACO: A bio-inspired modified version of the ant colony optimization algorithm to solve travel salesman problems

The Travel Salesman Problem (TSP) consists in finding the minimal-length closed tour that connects the entire group of nodes of a given graph. We propose to solve such a combinatorial optimization problem with the AddACO algorithm: it is a version of the Ant Colony Optimization method that is characterized by a modified probabilistic law at the basis of the exploratory movement of the artificial insects. In particular, the ant decisional rule is here set to amount in a linear convex combination of competing behavioral stimuli and has therefore an additive form (hence the name of our algorithm), rather than the canonical multiplicative one. The AddACO intends to address two conceptual shortcomings that characterize classical ACO methods: (i) the population of artificial insects is in principle allowed to simultaneously minimize/maximize all migratory guidance cues (which is in implausible from a biological/ecological point of view) and (ii) a given edge of the graph has a null probability to be explored if at least one of the movement trait is therein equal to zero, i.e., regardless the intensity of the others (this in principle reduces the exploratory potential of the ant colony). Three possible variants of our method are then specified: the AddACO-V1, which includes pheromone trail and visibility as insect decisional variables, and the AddACO-V2 and the AddACO-V3, which in turn add random effects and inertia, respectively, to the two classical migratory stimuli. The three versions of our algorithm are tested on benchmark middle-scale TPS instances, in order to assess their performance and to find their optimal parameter setting. The best performing variant is finally applied to large-scale TSPs, compared to the naive Ant-Cycle Ant System, proposed by Dorigo and colleagues, and evaluated in terms of quality of the solutions, computational time, and convergence speed. The aim is in fact to show that the proposed transition probability, as long as its conceptual advantages, is competitive from a performance perspective, i.e., if it does not reduce the exploratory capacity of the ant population w.r.t. the canonical one (at least in the case of selected TSPs). A theoretical study of the asymptotic behavior of the AddACO is given in the appendix of the work, whose conclusive section contains some hints for further improvements of our algorithm, also in the perspective of its application to other optimization problems

The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change