On the Complexity of the Multiple Stack TSP, kSTSP

Networks, 2013

This article studies the double traveling salesman problem with two stacks. A number of requests have to be served where each request consists in the pickup and delivery of an item. All the pickup operations have to be performed before any delivery can take place. A single vehicle is available that starts from a depot, performs all the pickup operations and returns to the depot. Then, it performs all the delivery operations and returns to the depot. The items are loaded in two stacks, each served independently from the other with a last-in-first-out policy. The objective is the minimization of the total cost of the pickup and delivery tours. We propose a branchand-bound approach to solve the problem. The algorithm uses properties of the problem both to tighten the lower bounds and to avoid the exploration of redundant subtrees. Computational results performed on benchmark instances reveal that the algorithm outperforms the other exact approaches for this problem.

Lecture Notes in Computer Science, 2012

In the uncapacitated asymmetric traveling salesman with multiple stacks, we perform a hamiltonian circuit to pick up n items, storing them in a vehicle with k stacks satisfying last-in-first-out constraints, and then we deliver every item by performing a hamiltonian circuit. We are interested in the convex hull of the (arc-)incidence vectors of such couples of hamiltonian circuits. For the general case, we determine the dimension of this polytope, and show that every facet of the asymmetric traveling salesman polytope defines one of its facets. For the special case with two stacks, we provide an integer linear programming formulation whose linear relaxation is polynomial-time solvable, and we propose new families of valid inequalities to reinforce this linear relaxation.

Theoretical Computer Science, 2014

In TSP with profits we have to find an optimal tour and a set of customers satisfying a specific constraint. In this paper we analyze three different variants of TSP with profits that are NP-hard in general. We study their computational complexity on special structures of the underlying graph, both in the case with and without service times to the customers. We present polynomial algorithms for the polynomially solvable cases and fully polynomial time approximation schemes (fptas) for some NP-hard cases.

Journal of the Operations Research Society of Japan

We collsider a generalization of the classical traveling salesman problem (TSP) called the precedence constrained traveling salesman problem (PCTSP), i.e. given a directed complete graph GCV,E), a distance Dij on each arc (i,j') E E, precedeilce constraints K oii V, we want te find a minimum distance tour that starts node 1 E V, visits all the nodes in V-{1}, and returns iiode 1 again so that node i i's vislted befbre node j' when i K)', We present a branch and bound algorithm for the exact solutions to the PCTSP incorporating lovver bounds computed from the Lagrangean relaxation. Our Iower bounding procedure is a generalization of the restricted Lagrangean method that lxas been successfully adapted to the TSP by Balas and Christofides [2]. Our branch and bound algoritlun also incorporates several heuristics and variable reduction tests. The computational results with up to 49 nodes show that our algorithm computes exact solutions to severa} classes of precedence constraints within acceptable cornputational requirements.

INFORMS Journal on Computing, 2013

The double traveling salesman problem with multiple stacks is a variant of the pickup and delivery traveling salesman problem in which all pickups must be completed before any delivery. In addition, items can be loaded on multiple stacks in the vehicle, and each stack must obey the last-in-first-out policy. The problem consists of finding the shortest Hamiltonian cycles covering all pickup and delivery locations while ensuring the feasibility of the loading plan. We formulate the problem as two traveling salesman problems linked by infeasible path constraints. We also introduce several strengthenings of these constraints, which are used within a branch-and-cut algorithm. Computational results performed on instances from the literature show that the algorithm outperforms existing exact algorithms. Instances with up to 28 requests (58 nodes) have been solved to optimality.

2002

Naval Research Logistics, 1999

We consider the Capacitated Traveling Salesman Problem with Pickups and Deliveries (CTSPPD). This problem is characterized by a set of n pickup points and a set of n delivery points. A single product is available at the pickup points which must be brought to the delivery points. A vehicle of limited capacity is available to perform this task. The problem is to determine the tour the vehicle should follow so that the total distance traveled is minimized, each load at a pickup point is picked up, each delivery point receives its shipment and the vehicle capacity is not violated. We present two polynomial-time approximation algorithms for this problem and analyze their worst-case bounds.

2003

In the Generalized Traveling Salesman Problem (GTSP), given a weighted complete digraph D and a partition V 1 ,. .. , V k of the vertices of D, we are to find a minimum weight cycle containing exactly one (at least one) vertex from each set V i , i = 1,. .. , k. Assignment Problem based approaches are extensively used for the Asymmetric TSP. To use analogs of these approaches for the GTSP, we need to find a minimum weight 1-regular subdigraph that contains exactly one (at least one) vertex from each V i. We prove that, unfortunately, the corresponding problems are NP-hard. In fact, we show the following stronger result: Let D = (V, A) be a digraph and let V 1 , V 2 ,. .. , V k be a partition of V. The problem of checking whether D has a 1-regular subdigraph containing exactly one vertex from each V 1 , V 2 ,. .. , V k is NP-complete even if |V i | ≤ 2 for every i = 1, 2,. .. , k.

We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum (or minimum) number of colors. We derive results regarding hardness of approximation, and analyze approximation algorithms for both versions of the problem. For the maximization version we give a 1 2 -approximation algorithm and show that it is APXhard. For the minimization version, we show that it is not approximable within n 1−ǫ for every ǫ > 0. When every color appears in the graph at most r times and r is an increasing function of n the problem is not O(r 1−ǫ )-approximable. For fixed constant r we analyze a polynomialtime (r + Hr)/2-approximation algorithm (Hr is the r-th harmonic number), and prove APX-hardness. Analysis of the studied algorithms is shown to be tight.

Discrete Applied Mathematics, 2009

Deciding whether or not a feasible solution to the Traveling Salesman Problem with Pickups and Deliveries (TSPPD) exists is polynomially solvable. We prove that counting the number of feasible solutions of the TSPPD is hard by showing the problem is #P-complete.