Budgeted Prize-Collecting Traveling Salesman and Minimum Spanning Tree Problems (original) (raw)

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Prize-Collecting TSP with a Budget Constraint

2017

We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint.

A primal-dual approximation algorithm for the Asymmetric Prize-Collecting TSP

Journal of Combinatorial Optimization, 2012

We present a primal-dual log(n)-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. The previous algorithm for the problem (V.H. Nguyen and T.T Nguyen in Int. J. Math. Oper. Res. 4(3):294-301, 2012) which is not combinatorial, is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.'s heuristic (Frieze et al. in Networks 12:23-39, 1982) or the recent Asadpour et al.'s heuristic for the ATSP (Asadpour et al. in 21st ACM-SIAM symposium on discrete algorithms, 2010). Depending on which of the two heuristics is used, it gives respectively 1 + log(n) and 3 + 8 log(n) log(log(n)) as an approximation ratio. Our algorithm achieves an approximation ratio of log(n) which is weaker than 3 + 8 log(n) log(log(n)) but represents the first combinatorial approximation algorithm for the Asymmetric Prize-Collecting TSP.

Prize Collecting Travelling Salesman Problem

Proceedings of 5th the International Conference on Operations Research and Enterprise Systems, 2016

The Prize Collecting Travelling Salesman Problem (PCTSP) is an important generalization of the famous Travelling Salesman Problem. It also arises as a sub problem in many variants of the Vehicle Routing Problem. In this paper, we provide efficient methods to solve the linear programming relaxation of the PCTSP. We provide efficient heuristics to obtain the Generalized Subtour Elimination Constraints (GSECs) for the PCTSP, and compare its performance with an optimal separation procedure. Furthermore, we show that a heuristic to separate the primitive comb inequalities for the TSP can be applied to separate the primitive comb inequalities introduced for the PCTSP. We evaluate the effectiveness of these inequalities in reducing the integrality gap for the PCTSP.

On Prize-collecting Tours and the Asymmetric Travelling Salesman Problem

International Transactions in Operational Research, 1995

We consider a variant of the Travelling Salesman Problem which is to determine a tour visiting each vertex in the graph at most at one time; if a vertex is left unrouted a given penalty has to be paid. The objective function is to find a balance between these psnalities and the cost of the tour. We call this problem the Profitable Tour Problem (PTP). If, in addition, each vertex is associated with a prize and there is a knapsack constraint which guarantees that a sufficiently large prize is collected, we have the well-known Prize-collecting Travelling Salesman Problem (PC'I~P). In this paper we summarize the main results presented in the literature, then we give lower bounds for the asymmetric version of PTP and PCTSP. Moreover, we show, through computational experiments, that large size instances of the asymmetric PTP can be solved exactly.

The online Prize-Collecting Traveling Salesman Problem

Information Processing Letters, 2008

We study the online version of the Prize-Collecting Traveling Salesman Problem (PCTSP), a generalization of the Traveling Salesman Problem (TSP). In the TSP, the salesman has to visit a set of cities while minimizing the length of the overall tour. In the PCTSP, each city has a given weight and penalty, and the goal is to collect a given quota of the weights of the cities while minimizing the length of the tour plus the penalties of the cities not in the tour. In the online version, cities are disclosed over time. We give a 7/3-competitive algorithm for the problem, which compares with a lower bound of 2 on the competitive ratio of any deterministic algorithm. We also show how our approach can be combined with an approximation algorithm in order to obtain an O(1)-competitive algorithm that runs in polynomial time.

Modeling and Solving the Traveling Salesman Problem with Priority Prizes

Pesquisa Operacional, 2018

This paper addresses the Traveling Salesman Problem with Priority Prizes (TSPPP), an extension of the classical TSP in which the order of the node visits is taken into account in the objective function. A prize p ki is received by the traveling salesman when node i is visited in the k-th order of the route, while a travel cost c i j is incurred when the salesman travels from node i to node j. The aim of the TSPPP is to find the maximum profit n-node tour. The problem can be seen as a TSP variant with a more general objective function, aiming at solutions that in some way consider the quality of customer service and the delivery priorities and costs. A natural representation for the TSPPP is here grounded in the point of view of Koopmans and Beckmann approach, according to which the problem is seem as a special case of the quadratic assignment problem (QAP). Given the novelty of this TSP variant, we propose different mixed integer programming models to appropriately represent the TSPPP, some of them based on the QAP. Computational experiments are also presented when solving the MIP models with a well-known optimization software, as well as with a tabu search algorithm.

Primal-dual approximation algorithms for the Prize-Collecting Steiner Tree Problem

Information Processing Letters, 2007

The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2 − 1 n−1 )-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n 3 log n) time. Their algorithm applies the primal-dual scheme once for each of the n vertices of the graph. We present a primal-dual algorithm that runs in O(n 2 log n), as it applies this scheme only once, and achieves the slightly better ratio of 2 − 2 n . We also show a tight example for the analysis of the algorithm and discuss briefly a couple of other algorithms described in the literature.

A branch-and-cut and MIP-based heuristics for the Prize-Collecting Travelling Salesman Problem

RAIRO - Operations Research

The Prize Collecting Traveling Salesman Problem (PCTSP) represents a generalization of the well-known Traveling Salesman Problem. The PCTSP can be associated with a salesman that collects a prize in each visited city and pays a penalty for each unvisited city, with travel costs among the cities. The objective is to minimize the sum of the costs of the tour and penalties, while collecting a minimum amount of prize. This paper suggests MIP-based heuristics and a branch-and-cut algorithm to solve the PCTSP. Experiments were conducted with instances of the literature, and the results of our methods turned out to be quite satisfactory.

A Tabu Search Approach for the Prize Collecting Traveling Salesman Problem

Electronic Notes in Discrete Mathematics, 2013

The Prize Collecting Traveling Salesman Problem is a generalization of the Traveling Salesman Problem. A salesman collects a prize for each visited city and pays a penalty for each non visited city. The objective is to minimize the sum of the travel costs and penalties, but collecting a minimum pre-established amount of prizes. This problem is here addressed by a simple, but efficient tabu search approach which had improved several upper bounds of the considered instances.

Hybrid Metaheuristic for the Prize Collecting Travelling Salesman Problem

Lecture Notes in Computer Science, 2008

The Prize Collecting Traveling Salesman Problem (PCTSP) can be associated to a salesman that collects a prize in each city visited and pays a penalty for each city not visited, with travel costs among the cities. The objective is to minimize the sum of travel costs and penalties, while including in the tour enough cities to collect a minimum prize. This paper presents one solution procedure for the PCTSP, using a hybrid metaheuristic known as Clustering Search (CS), whose main idea is to identify promising areas of the search space by generating solutions and clustering them into groups that are them explored further. The validation of the obtained solutions was through the comparison with the results found by CPLEX.