506 research outputs found
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 with edges and vertices, and , our
randomized algorithm outputs with high probability a -approximate
solution to the LP relaxation whose support has edges. The running time of the algorithm is
. 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
with edges and vertices, and , we describe an
algorithm that outputs with high probability a tour of with cost at most
times the minimum cost tour of in time
. Previous
implementations of Christofides' algorithm [Christofides, 1976] require, for a
-optimal tour, time when the metric
is explicitly given, or 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
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