An Experimental Evaluation of the Best-of-Many Christofides' Algorithm for the Traveling Salesman Problem

Recent papers on approximation algorithms for the traveling salesman problem (TSP) have given a new variant on the well-known Christofides' algorithm for the TSP, called the Best-of-Many Christofides' algorithm. The algorithm involves sampling a spanning tree from the solution the standard LP relaxation of the TSP, subject to the condition that each edge is sampled with probability at most its value in the LP relaxation. One then runs Christofides' algorithm on the tree by computing a minimum-cost matching on the odd-degree vertices in the tree, and shortcutting the resulting Eulerian graph to a tour. In this paper we perform an experimental evaluation of the Best-of-Many Christofides' algorithm to see if there are empirical reasons to believe its performance is better than that of Christofides' algorithm. Furthermore, several different sampling schemes have been proposed; we implement several different schemes to determine which ones might be the most promising for obtaining improved performance guarantees over that of Christofides' algorithm. In our experiments, all of the implemented methods perform significantly better than the Christofides' algorithm; an algorithm that samples from a maximum entropy distribution over spanning trees seems to be particularly good, though there are others that perform almost as well.Comment: An extended abstract of this paper will appear in ESA 201

Fast Approximations for Metric-TSP via Linear Programming

We develop faster approximation algorithms for Metric-TSP building on recent, nearly linear time approximation schemes for the LP relaxation [Chekuri and Quanrud, 2017]. We show that the LP solution can be sparsified via cut-sparsification techniques such as those of Benczur and Karger [2015]. Given a weighted graph $G$ with $m$ edges and $n$ vertices, and $\epsilon > 0$, our randomized algorithm outputs with high probability a $(1+\epsilon)$-approximate solution to the LP relaxation whose support has $\operatorname{O}(n \log n /\epsilon^2)$ edges. The running time of the algorithm is $\operatorname{\~O}(m/\epsilon^2)$. This can be generically used to speed up algorithms that rely on the LP. For Metric-TSP, we obtain the following concrete result. For a weighted graph $G$ with $m$ edges and $n$ vertices, and $\epsilon > 0$, we describe an algorithm that outputs with high probability a tour of $G$ with cost at most $(1 + \epsilon) \frac{3}{2}$ times the minimum cost tour of $G$ in time $\operatorname{\~O}(m/\epsilon^2 + n^{1.5}/\epsilon^3)$. Previous implementations of Christofides' algorithm [Christofides, 1976] require, for a $\frac{3}{2}$-optimal tour, $\operatorname{\~O}(n^{2.5})$ time when the metric is explicitly given, or $\operatorname{\~O}(\min\{m^{1.5}, mn+n^{2.5}\})$ time when the metric is given implicitly as the shortest path metric of a weighted graph

Reassembling trees for the traveling salesman

Many recent approximation algorithms for different variants of the traveling salesman problem (asymmetric TSP, graph TSP, s-t-path TSP) exploit the well-known fact that a solution of the natural linear programming relaxation can be written as convex combination of spanning trees. The main argument then is that randomly sampling a tree from such a distribution and then completing the tree to a tour at minimum cost yields a better approximation guarantee than simply taking a minimum cost spanning tree (as in Christofides' algorithm). We argue that an additional step can help: reassembling the spanning trees before sampling. Exchanging two edges in a pair of spanning trees can improve their properties under certain conditions. We demonstrate the usefulness for the metric s-t-path TSP by devising a deterministic polynomial-time algorithm that improves on Seb\H{o}'s previously best approximation ratio of 8/5.Comment: minor revision, final version, to appear in SIAM Journal of Discrete Mathematics, please use color printe

Learning Combinatorial Optimization Algorithms over Graphs

The design of good heuristics or approximation algorithms for NP-hard combinatorial optimization problems often requires significant specialized knowledge and trial-and-error. Can we automate this challenging, tedious process, and learn the algorithms instead? In many real-world applications, it is typically the case that the same optimization problem is solved again and again on a regular basis, maintaining the same problem structure but differing in the data. This provides an opportunity for learning heuristic algorithms that exploit the structure of such recurring problems. In this paper, we propose a unique combination of reinforcement learning and graph embedding to address this challenge. The learned greedy policy behaves like a meta-algorithm that incrementally constructs a solution, and the action is determined by the output of a graph embedding network capturing the current state of the solution. We show that our framework can be applied to a diverse range of optimization problems over graphs, and learns effective algorithms for the Minimum Vertex Cover, Maximum Cut and Traveling Salesman problems.Comment: NIPS 201

Chaotic Simulated Annealing by A Neural Network Model with Transient Chaos

We propose a neural network model with transient chaos, or a transiently chaotic neural network (TCNN) as an approximation method for combinatorial optimization problem, by introducing transiently chaotic dynamics into neural networks. Unlike conventional neural networks only with point attractors, the proposed neural network has richer and more flexible dynamics, so that it can be expected to have higher ability of searching for globally optimal or near-optimal solutions. A significant property of this model is that the chaotic neurodynamics is temporarily generated for searching and self-organizing, and eventually vanishes with autonomous decreasing of a bifurcation parameter corresponding to the "temperature" in usual annealing process. Therefore, the neural network gradually approaches, through the transient chaos, to dynamical structure similar to such conventional models as the Hopfield neural network which converges to a stable equilibrium point. Since the optimization process of the transiently chaotic neural network is similar to simulated annealing, not in a stochastic way but in a deterministically chaotic way, the new method is regarded as chaotic simulated annealing (CSA). Fundamental characteristics of the transiently chaotic neurodynamics are numerically investigated with examples of a single neuron model and the Traveling Salesman Problem (TSP). Moreover, a maintenance scheduling problem for generators in a practical power system is also analysed to verify practical efficiency of this new method.Comment: the theoretical results related to this paper should be referred to "Chaos and Asymptotical Stability in Discrete-time Neural Networks" by L.Chen and K.Aihara, Physica D (in press). Journal ref.: Neural Networks, Vol.8, No.6, pp.915-930, 199

An Approximation Approach for Solving the Subpath Planning Problem

The subpath planning problem is a branch of the path planning problem, which has widespread applications in automated manufacturing process as well as vehicle and robot navigation. This problem is to find the shortest path or tour subject for travelling a set of given subpaths. The current approaches for dealing with the subpath planning problem are all based on meta-heuristic approaches. It is well-known that meta-heuristic based approaches have several deficiencies. To address them, we propose a novel approximation algorithm in the O(n^3) time complexity class, which guarantees to solve any subpath planning problem instance with the fixed ratio bound of 2. Also, the formal proofs of the claims, our empirical evaluation shows that our approximation method acts much better than a state-of-the-art method, both in result and execution time

Anytime Behavior of Inexact TSP Solvers and Perspectives for Automated Algorithm Selection

The Traveling-Salesperson-Problem (TSP) is arguably one of the best-known NP-hard combinatorial optimization problems. The two sophisticated heuristic solvers LKH and EAX and respective (restart) variants manage to calculate close-to optimal or even optimal solutions, also for large instances with several thousand nodes in reasonable time. In this work we extend existing benchmarking studies by addressing anytime behaviour of inexact TSP solvers based on empirical runtime distributions leading to an increased understanding of solver behaviour and the respective relation to problem hardness. It turns out that performance ranking of solvers is highly dependent on the focused approximation quality. Insights on intersection points of performances offer huge potential for the construction of hybridized solvers depending on instance features. Moreover, instance features tailored to anytime performance and corresponding performance indicators will highly improve automated algorithm selection models by including comprehensive information on solver quality.Comment: This version has been accepted for publication at the IEEE Congress on Evolutionary Computation (IEEE CEC) 2020, which is part of the IEEE World Congress on Computational Intelligence (IEEE WCCI) 202

From Symmetry to Asymmetry: Generalizing TSP Approximations by Parametrization

We generalize the tree doubling and Christofides algorithm, the two most common approximations for TSP, to parameterized approximations for ATSP. The parameters we consider for the respective parameterizations are upper bounded by the number of asymmetric distances in the given instance, which yields algorithms to efficiently compute constant factor approximations also for moderately asymmetric TSP instances. As generalization of the Christofides algorithm, we derive a parameterized 2.5-approximation, where the parameter is the size of a vertex cover for the subgraph induced by the asymmetric edges. Our generalization of the tree doubling algorithm gives a parameterized 3-approximation, where the parameter is the number of asymmetric edges in a given minimum spanning arborescence. Both algorithms are also stated in the form of additive lossy kernelizations, which allows to combine them with known polynomial time approximations for ATSP. Further, we combine them with a notion of symmetry relaxation which allows to trade approximation guarantee for runtime. We complement our results by experimental evaluations, which show that generalized tree-doubling frequently outperforms generalized Christofides with respect to parameter size

A Comparative Study of Adaptive Crossover Operators for Genetic Algorithms to Resolve the Traveling Salesman Problem

Genetic algorithm includes some parameters that should be adjusting so that the algorithm can provide positive results. Crossover operators play very important role by constructing competitive Genetic Algorithms (GAs). In this paper, the basic conceptual features and specific characteristics of various crossover operators in the context of the Traveling Salesman Problem (TSP) are discussed. The results of experimental comparison of more than six different crossover operators for the TSP are presented. The experiment results show that OX operator enables to achieve a better solutions than other operators tested

Analyzing the Performance of Mutation Operators to Solve the Travelling Salesman Problem

The genetic algorithm includes some parameters that should be adjusted, so as to get reliable results. Choosing a representation of the problem addressed, an initial population, a method of selection, a crossover operator, mutation operator, the probabilities of crossover and mutation, and the insertion method creates a variant of genetic algorithms. Our work is part of the answer to this perspective to find a solution for this combinatorial problem. What are the best parameters to select for a genetic algorithm that creates a variety efficient to solve the Travelling Salesman Problem (TSP)? In this paper, we present a comparative analysis of different mutation operators, surrounded by a dilated discussion that justifying the relevance of genetic operators chosen to solving the TSP problem.Comment: ISSN: 2222-425