Variant Of Traveling Salesman Problem Research Articles

An Efficient Hybrid Genetic Algorithm for the Quadratic Traveling Salesman Problem

The traveling salesman problem (TSP) is the most well-known problem in combinatorial optimization which has been studied for many decades. This paper focuses on dealing with one of the most difficult TSP variants named the quadratic traveling salesman problem (QTSP) that has numerous planning applications in robotics and bioinformatics. The goal of QTSP is similar to TSP which finds a cycle visiting all nodes exactly once with minimum total costs. However, the costs in QTSP are associated with three vertices traversed in succession (instead of two like in TSP). This leads to a quadratic objective function that is much harder to solve. To efficiently solve the problem, we propose a hybrid genetic algorithm including a local search procedure for intensification and a new mutation operator for diversification. The local search is composed of a restricted double-bridge move (a variant of 4-Opt); and we show the neighborhood can be evaluated in O(n^2), the same complexity as for the classical TSP. The mutation phase is inspired by a ruin-and-recreate scheme. Experimental results conducted on benchmark instances show that our method significantly outperforms state-of-the-art algorithms in terms of solution quality. Out of 800 considered instances, it finds 437 new best-known solutions.

Reinforced Lin–Kernighan–Helsgaun algorithms for the traveling salesman problems

The Traveling Salesman Problem (TSP) is a classical NP-hard combinatorial optimization problem with many practical variants. The Lin–Kernighan–Helsgaun (LKH) algorithm is one of the state-of-the-art local search algorithms for the TSP. LKH-3 is a powerful extension of LKH that can solve many TSP variants. Both LKH and LKH-3 use the candidate sets to improve their efficiency, and have two different methods, α-measure and POPMUSIC, to decide the candidate sets. In this work, we first propose a Variable Strategy Reinforced LKH (VSR-LKH) algorithm, which incorporates three reinforcement learning methods (Q-learning, Sarsa, and Monte Carlo) with LKH, for the TSP. We further propose a new algorithm called VSR-LKH-3 that combines the variable strategy reinforcement learning method with LKH-3 for typical TSP variants, including the TSP with Time Windows (TSPTW) and the Colored TSP (CTSP). The proposed algorithms replace the inflexible traversal operations in LKH and LKH-3 and let the algorithms learn to make a choice at each search step by reinforcement learning. Both LKH and LKH-3, with either the α-measure or the POPMUSIC method, can be significantly improved by our methods. Extensive experiments on 236 widely-used TSP benchmarks with up to 85,900 cities demonstrate the excellent performance of VSR-LKH. VSR-LKH-3 also significantly outperforms the state-of-the-art heuristics for the TSPTW and CTSP.

ITÖ algorithm with local search for large scale multiple balanced traveling salesmen problem

In the fields such as computer science, intelligent transport systems and logistics scheduling, many problems can be reflected as the variants of traveling salesman problem (TSP). In recent years, although there are many achievements and developments on the study of intelligent optimization algorithms for TSP and its variants, the challenge and problem are still existed: while the scale of problem is large, there is the dimension disaster problem; the optimization algorithms for TSP variants are easy to fall into the local optimum. Multiple balanced traveling salesmen problem (MBTSP) is a variant of TSP, it can be used in the fields such as optimizing multiple gas turbine engines. Since MBTSP is NP-hard problem, many intelligent optimization algorithms, such as genetic algorithm and ITÖ algorithm, have been used to solve it. However, while the scale of MBTSP is large, the traditional algorithms are easy to fall into local optimum. Aiming at the problem, this paper extends the scale of MBTSP to large scale scenes, and proposes a novel ITÖ algorithm (NITÖ) based on crossover operator and local search for large scale MBTSP. For NITÖ algorithm, the drift operator and volatility operator of ITÖ algorithm are redesigned, which are carried out by improved crossover operator and local search. The extensive experiments show that NITÖ can demonstrate better solution quality than the compared state-of-the-art algorithms.

An ils-based heuristic applied to the car renter salesman problem

The present paper tackles the Car Renter Salesman Problem (CaRS), which is a Traveling Salesman Problem variant. In CaRS, the goal is to travel through a set of cities using rented vehicles at minimum cost. The main aim of the current problem is to establish an optimal route using rented vehicles of different types to each trip. Since CaRS is N P-Hard, we herein present a heuristic approach to tackle it. The approach is based on a Multi-Start Iterated Local Search metaheuristic, where the local search step is based on the Random Variable Neighborhood Descent methodology. An Integer Linear Programming Formulation based on a Quadratic Formulation from literature is also proposed in the current study. Computational results for the proposed heuristic method in euclidean instances outperform current state-of-the-art results. The proposed formulation also has stronger bounds and relaxation when compared to others from literature.

Fine-grained Complexity Analysis of Two Classic TSP Variants

We analyze two classic variants of the T RAVELING S ALESMAN P ROBLEM ( TSP ) using the toolkit of fine-grained complexity. Our first set of results is motivated by the B ITONIC TSP problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in O ( n 2 ) time. While the near-quadratic dependency of similar dynamic programs for L ONGEST C OMMON S UBSEQUENCE and D ISCRETE F réchet D istance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in O ( n log 2 n ) time and its bottleneck version in O ( n log 3 n ) time. In the more general pyramidal TSP problem, the points to be visited are labeled 1,… , n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) TSP problem also work for the pyramidal TSP problem in the plane. Our second set of results concerns the popular k - OPT heuristic for TSP in the graph setting. More precisely, we study the k - OPT decision problem, which asks whether a given tour can be improved by a k - OPT move that replaces k edges in the tour by k new edges. A simple algorithm solves k - OPT in O ( n k ) time for fixed k . For 2- OPT , this is easily seen to be optimal. For k =3, we prove that an algorithm with a runtime of the form Õ( n 3−ɛ ) exists if and only if A LL -P AIRS S HORTEST P ATHS in weighted digraphs has such an algorithm. For general k - OPT , it is known that a runtime of f ( k ) · n o ( k / log k ) would contradict the Exponential Time Hypothesis. The results for k =2,3 may suggest that the actual time complexity of k - OPT is Θ ( n k ). We show that this is not the case, by presenting an algorithm that finds the best k -move in O ( n ⌊ 2 k /3 ⌋+1 ) time for fixed k ≥ 3. This implies that 4- OPT can be solved in O ( n 3 ) time, matching the best-known algorithm for 3- OPT . Finally, we show how to beat the quadratic barrier for k =2 in two important settings, namely, for points in the plane and when we want to solve 2- OPT repeatedly.

빅데이터를 활용한 다구간(多區間) 항공 경로 최적화 시스템

This paper proposes a multi-city flight route optimization system using big-data analytics. When one plans a travel for visiting a set of cities, the sequence of visiting should be determined so the total travel cost is minimized. Such problem is known as the travelling salesman problem (TSP) in the literature. We present a variant of TSP considering the dynamic travel costs between two cities and a total travel duration. We present a mixed integer programming formulation for solving the problem, with developing a flight price trend prediction model. The goal of the flight price trend prediction model is to reduce time and cost that are required to construct the route optimization problem. We employed big-data analytics to build the appropriate price trend prediction model that involves massive web crawling, statistical analysis, and regression. A web-based system was, then, built on the developed models for demonstrating the validity of the proposed approach.

New Formulations for the Orienteering Problem

Abstract Problems associated with determining optimal routes from one or several depots (origin, home city) to a set of nodes (vertices, cities, customers, locations) are known as routing problems. The Traveling Salesman Problem (TSP) lies at the heart of routing problems. One of the new variants of the TSP is named as TSP with Profits where the traveler must finish its journey within a predetermined time (cost, distance), by optimizing given objective. In this variant of TSP, all cities ought to not to be visited. The Orienteering Problem (OP) is the most studied case of TSP with Profits which comes from an outdoor sport played on mountains. In OP, traveler gets a gain (profit, reward) from the visited node and the objective is to maximize the total gain that the traveler collects during the predetermined time. The OP is also named as selective TSP. In this paper, we present two polynomial size formulations for OP. The performance of our proposed formulations is tested on benchmark instances. We solved the benchmark problems from the literature via CPLEX 12.5 by using the proposed formulations and existing formulation. The computational experiments demonstrate that; (1) both of the new formulations over estimates the existing one; and (2) the proposed formulations are capable of solving all the benchmark instances that were solved by using special heuristics so far.

The Power of Recourse for Online MST and TSP

We consider online versions of the minimum spanning tree (MST) problem and the traveling salesman problem (TSP) where recourse is allowed. The nodes of an unknown graph with metric edge cost appear one by one and must be connected in such a way that the resulting tree or tour has low cost. In the standard online setting, with irrevocable decisions, no algorithm can guarantee a constant-competitive ratio. In our model we allow recourse actions by giving a limited budget of edge rearrangements per iteration. It has been an open question for more than 20 years whether an online algorithm equipped with a constant (amortized) budget can guarantee constant-approximate solutions. As our main result, we answer this question affirmatively in an amortized setting. We introduce an algorithm that maintains a nearly optimal tree when given a constant amortized budget. Unlike in classical TSP variants, the standard double-tree and shortcutting approach does not give constant guarantees in the online setting. We propose...

A 3/2-Approximation Algorithm for the Multiple TSP with a Fixed Number of Depots

We study a natural extension of the classical traveling salesman problem (TSP) in the situation where multiple salesmen are dispatched from a number of different depots. As with the TSP, this problem is motivated by a large range of applications in vehicle routing. Although it is known to have a 2-approximation algorithm, whether the problem has a 3/2-approximation algorithm, as is the case with the well-known Christofides heuristic for the TSP, remains an open question. We answer this question positively by providing a 3/2-approximation algorithm for the problem with a fixed number of depots. The algorithm uses an edge exchange strategy, and its analysis hinges on a newly discovered exchange property of matroids. In addition, the algorithm is applied to multidepot extensions of other TSP variants, and we show for the first time, to our knowledge, that for these multidepot extensions the same best constant approximation ratios can be achieved as for their respective single-depot cases.

An application of the self-organizing map in the non-Euclidean Traveling Salesman Problem

An application of the self-organizing map (SOM) to the Traveling Salesman Problem (TSP) has been reported by many researchers, however these approaches are mainly focused on the Euclidean TSP variant. We consider the TSP as a problem formulation for the multi-goal path planning problem in which paths among obstacles have to be found. We apply a simple approximation of the shortest path that seems to be suitable for the SOM adaptation procedure. The approximation is based on a geometrical interpretation of SOM, where weights of neurons represent nodes that are placed in the polygonal domain. The approximation is verified in a set of real problems and experimental results show feasibility of the proposed approach for the SOM based solution of the non-Euclidean TSP.

TSP in spreadsheets—A fast and flexible tool

The traveling salesman problem (TSP) is well-known and many specially developed solution procedures have been constructed to solve particular variants of it. This paper considers several different variants of TSP. However, developing tailored solution procedures for each is impractical. These problems are non-deterministic polynomial-time hard (NP hard). Solving them using standard linear programming/mixed integer programming (LP/MIP) solvers has therefore only been regarded to be feasible for very small problems. A careful consideration of the problem formulation may facilitate efficient software utilization, and for real-world problems this can have a considerable impact. Problems that were previously regarded as large and unwieldy are now easily solvable using spreadsheets, thanks to the recent advancement in general optimization software. A comparison of spreadsheet solvers is made with other general purpose optimization tools like the Cplex solver. Here, spreadsheet solvers compete very well. Alternative formulations are implemented that are capable of solving real-world TSP variants that are not suitable for specialized solvers tailored to standard TSP. A performance evaluation is also made between a standard “tight” formulation and a general “wide” formulation. It is found that some solvers are more sensitive to the type of formulation than others.

The moving-target traveling salesman problem

Previous literature on the Traveling Salesman Problem (TSP) assumed that the sites to be visited are stationary. Motivated by practical applications, we introduce a time-dependent generalization of TSP which we call Moving-Target TSP, where a pursuer must intercept in minimum time a set of targets which move with constant velocities. We propose approximate and exact algorithms for several natural variants of Moving-Target TSP.

Simple Algorithms for Gilmore–Gomory's Traveling Salesman and Related Problems

We reconsider the version of the traveling salesman problem (TSP) first studied in a well-known paper by Gilmore and Gomory (1964). In this, the distance between two cities A and B, is an integrable function of the x-coordinate of A and the y-coordinate of B. This problem finds important applications in machine scheduling, workforce planning, and combinatorial optimization. We solve this TSP variant by a {\mathcal O}(n log n) algorithm considerably simpler than previously known algorithms. The new algorithm demonstrates and exploits the structure of an optimal solution, and recreates it using minimal storage space without the use of edge interchanges.

Approximation Results for Kinetic Variants of TSP

We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem (TSP) where we consider instances in which each point moves with a fixed constant speed in a fixed direction. We prove the following results: • If the points all move with the same velocity, then there is a polynomial time approximation scheme for the Kinetic TSP. • The Kinetic TSP cannot be approximated better than by a factor of 2 by a polynomial time algorithm unless P = NP, even if there are only two moving points in the instance. • The Kinetic TSP cannot be approximated better than by a factor of $2^{\Omega(\sqrt{n})}$ by a polynomial time algorithm unless P = NP, even if the maximum velocity is bounded. n denotes the size of the input instance. The last result is especially surprising in the light of existing polynomial time approximation schemes for the static version of the problem.