Improvements to the Or-opt Heuristic for the Symmetric Traveling Salesman Problem
01 Jan 2005-pp 1-14
TL;DR: Several variants and generalizations of the Or-opt heuristic for the Symmetric Travelling Salesman Problem are developed and compared on random and planar instances to significantly improve upon the standard 2-opt and Or- opt heuristics.
Abstract: Several variants and generalizations of the Or-opt heuristic for the Symmetric Travelling Salesman Problem are developed and compared on random and planar instances. Some of the proposed algorithms are shown to significantly improve upon the standard 2-opt and Or-opt heuristics.
Citations
More filters
01 Jan 2017
TL;DR: This work proposes a global ranking constraint and shows that GAC can be achieved in polynomial time and proposes an Oðn3 log nÞ algorithm for achieving RC as well as an efficient quadratic algorithm offering a better tradeoff.
Abstract: In many problems we want to reason about the ranking of items. For example, in information retrieval, when aggregating several search results, we may have ties and consequently rank orders. (e.g. [2, 3]). As a second example, we may wish to construct an overall ranking of tennis player based on pairwise comparisons between players. One principled method for constructing a ranking is the Kemeny distance [5] as this is the unique scheme that is neutral, consistent, and Condorcet. Unfortunately, determining this ranking is NP-hard, and remains so when we permit ties in the input or output [4]. As a third example, tasks in a scheduling problem may run in parallel, resulting in a ranking. In a ranking, unlike a permutation, we can have ties. Thus, 12225 is a ranking whilst 12345 is a permutation. To reason about permutations, we have efficient and effective global constraints. Regin [7] proposed an Oðn4Þ GAC propagator for permutations. For BC, there is an even faster Oðn log nÞ propagator [6]. Every constraint toolkit now provides propagators for permutation constraints. Surprisingly, ranking constraints are not yet supported. In [1], we tackle this weakness by proposing a global ranking constraint. We show that simple decompositions of this constraint hurt pruning. We then show that GAC can be achieved in polynomial time and we propose an Oðn3 log nÞ algorithm for achieving RC as well as an efficient quadratic algorithm offering a better tradeoff.
113 citations
Cites background or methods from "Improvements to the Or-opt Heuristi..."
...Table 1 gives the values of the four characteristics of regular expressions for some regular expressions of [5], while Tables 2 and 3 provide the intervals of interest for 12 time-series constraints....
[...]
...Table 3 gives for 6 regular expressions of [5] the corresponding intervals of interest of sum_surf_σ constraints wrt some integer interval domain [ , u] such that u > 1 ∧ u− > 1, as well as the lower bound LB on the parameter N of the derived among constraint for time series that may have at least one σ-pattern....
[...]
...While coming up with implied constraints is usually problem specific, the theoretical contribution of this paper is a unique per family among implied constraint, that is valid for all regular expressions of the time-series constraint catalogue [5] and that covers all the 22 time-series constraints of the corresponding family....
[...]
...One principled method for constructing a ranking is the Kemeny distance [5] as this is the unique scheme that is neutral, consistent, and Condorcet....
[...]
...Table 2 gives for 6 regular expressions of [5] the corresponding intervals of interest of max_surf_σ constraints wrt some integer interval domain [ , u] such that u > 1 ∧ u− > 1, as well as the lower bound LB on the parameter N of the derived among constraint for time series that may have at least one σ-pattern....
[...]
01 Oct 2009
TL;DR: The Traveling Salesman Problem (TSP) and the Vehicle Routing Problem (VRP) are two of the most popular problems in the field of combinatorial optimization.
Abstract: The Traveling Salesman Problem (TSP) and the Vehicle Routing Problem (VRP) are two of the most popular problems in the field of combinatorial optimization. Due to the study of these two problems, there has been a significant growth in families of exact and heuristic algorithms being used today. The purpose of this paper is to show how their study has fostered developments of the most popular algorithms now applied to the solution of combinatorial optimization problems. These include exact algorithms, classical heuristics and metaheuristics.
13 citations
Journal Article•
10 citations
TL;DR: Wang et al. as mentioned in this paper extended a novel transportation model for the waste management system, namely, capacitated location routing problem with queuing time (CLRPQT), and designed a cross-entropy and simulated-annealing based hyper-heuristic algorithm (CE-SAHH) for it.
Abstract: Waste sorting is an imperative and significant issue in China, of which sorted-waste collection and transportation are indispensable parts. Despite its vital yet practical significance, few studies research mathematical models or algorithms of waste collection and transportation from the perspective of waste sorting. To address this issue, we extend a novel transportation model for the waste management system, namely, capacitated location routing problem with queuing time (CLRPQT) and design a cross-entropy and simulated-annealing based hyper-heuristic algorithm (CE-SAHH) for it. The main idea of this paper is three-fold: (1) As a particular property of this problem, source nodes cannot but need to be served by more than one vehicle that causes queuing time between a heterogeneous fleet of vehicles, which is novel in terms of the proposed model; (2) For the methodological contribution, a character encoding scheme, new decoding procedure, and local search strategy are designed embedded in the proposed method; (3) An integration of simulated annealing strategy and the cross-entropy-based hyper-heuristic algorithm is developed to overcome the combinatorial optimization problem with a more complex solution of this study. Finally, the results and analysis of three numeric experiments on benchmark datasets, new instances of CLRPQT, and simulation data in Shanghai, China, verify the effectiveness and universality of the proposed model and method.
2 citations
References
More filters
TL;DR: This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-salesman problem based on a general approach to heuristics that is believed to have wide applicability in combinatorial optimization problems.
Abstract: This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-salesman problem. The procedure is based on a general approach to heuristics that is believed to have wide applicability in combinatorial optimization problems. The procedure produces optimum solutions for all problems tested, "classical" problems appearing in the literature, as well as randomly generated test problems, up to 110 cities. Run times grow approximately as n2; in absolute terms, a typical 100-city problem requires less than 25 seconds for one case GE635, and about three minutes to obtain the optimum with above 95 per cent confidence.
3,761 citations
"Improvements to the Or-opt Heuristi..." refers methods in this paper
...Sophisticated implementations (for example, Johnson and McGeoch (1997, 2002) and Helsgaun (2000)) of the dynamic r-opt heuristic, in which the value of r is allowed to vary during the search (Lin and Kernighan, 1973), probably constitute the best available heuristics for the STSP....
[...]
01 Jan 2001
TL;DR: In this paper, the authors present a comprehensive overview of the most important techniques proposed for the solution of hard combinatorial problems in the area of vehicle routing problems, focusing on a specific family of problems.
Abstract: The Vehicle Routing Problem covers both exact and heuristic methods developed for the VRP and some of its main variants, emphasizing the practical issues common to VRP. The book is composed of three parts containing contributions from well-known experts. The first part covers basic VRP, known more commonly as capacitated VRP. The second part covers three main variants of VRP with time windows, backhauls, and pickup and delivery. The third part covers issues arising in real-world VRP applications and includes both case studies and references to software packages. The book will be of interest to both researchers and graduate-level students in the communities of operations research and matematical sciences. It focuses on a specific family of problems while offering a complete overview of the effective use of the most important techniques proposed for the solution of hard combinatorial problems. Practitioners will find this book particularly usef
3,395 citations
TL;DR: Two algorithms for solving the (symmetric distance) traveling salesman problem have been programmed for a high-speed digital computer and are based on a general heuristic approach believed to be of general applicability to various optimization problems.
Abstract: Two algorithms for solving the (symmetric distance) traveling salesman problem have been programmed for a high-speed digital computer. The first produces guaranteed optimal solution for problems involving no more than 13 cities; the time required (IBM 7094 II) varies from 60 milliseconds for a 9-city problem to 1.75 seconds for a 13-city problem. The second algorithm produces precisely characterized, locally optimal solutions for large problems (up to 145 cities) in an extremely short time and is based on a general heuristic approach believed to be of general applicability to various optimization problems. The average time required to obtain a locally optimal solution is under 30n3 microseconds where n is the number of cities involved. Repeated runs on a problem from random initial tours result in a high probability of finding the optimal solution among the locally optimal solutions obtained. For large problems where many locally optimal solutions have to be obtained in order to be reasonably assured of having the optimal solution, an efficient reduction scheme is incorporated in the program to reduce the total computation time by a substantial amount.
1,946 citations
"Improvements to the Or-opt Heuristi..." refers methods in this paper
...The generalized concept (Lin, 1965) extends the basic 2-opt method (Croes, 1958)....
[...]
Book•
01 Apr 1997
TL;DR: Local search is still the method of choice for NP-hard problems as it provides a robust approach for obtaining high-quality solutions to problems of a realistic size in a reasonable time.
Abstract: From the Publisher:
In the past three decades local search has grown from a simple heuristic idea into a mature field of research in combinatorial optimization. Local search is still the method of choice for NP-hard problems as it provides a robust approach for obtaining high-quality solutions to problems of a realistic size in a reasonable time. This area of discrete mathematics is of great practical use and is attracting ever increasing attention. The contributions to this book cover local search and its variants from both a theoretical and practical point of view, each with a chapter written by leading authorities on that particular aspect. This book is an important reference volume and an invaluable source of inspiration for advanced students and researchers in discrete mathematics, computer science, operations research, industrial engineering and management science.
1,901 citations
TL;DR: An implementation of the Lin–Kernighan heuristic, one of the most successful methods for generating optimal or near-optimal solutions for the symmetric traveling salesman problem (TSP), is described.
1,462 citations