813 research outputs found
Improving Christofides' Algorithm for the s-t Path TSP
We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t
path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices
including two prespecified endpoints, the problem is to find a shortest
Hamiltonian path between the two endpoints; Hoogeveen showed that the natural
variant of Christofides' algorithm is a 5/3-approximation algorithm for this
problem, and this asymptotically tight bound in fact has been the best
approximation ratio known until now. We modify this algorithm so that it
chooses the initial spanning tree based on an optimal solution to the Held-Karp
relaxation rather than a minimum spanning tree; we prove this simple but
crucial modification leads to an improved approximation ratio, surpassing the
20-year-old barrier set by the natural Christofides' algorithm variant. Our
algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of
the path-variant Held-Karp relaxation. The techniques devised in this paper can
be applied to other optimization problems as well: these applications include
improved approximation algorithms and improved LP integrality gap upper bounds
for the prize-collecting s-t path problem and the unit-weight graphical metric
s-t path TSP.Comment: 31 pages, 5 figure
A 3/2-Approximation for the Metric Many-visits Path TSP
In the Many-visits Path TSP, we are given a set of cities along with
their pairwise distances (or cost) , and moreover each city comes
with an associated positive integer request .
The goal is to find a minimum-cost path, starting at city and ending at
city , that visits each city exactly times.
We present a -approximation algorithm for the metric Many-visits
Path TSP, that runs in time polynomial in and poly-logarithmic in the
requests .
Our algorithm can be seen as a far-reaching generalization of the
-approximation algorithm for Path TSP by Zenklusen (SODA 2019), which
answered a long-standing open problem by providing an efficient algorithm which
matches the approximation guarantee of Christofides' algorithm from 1976 for
metric TSP.
One of the key components of our approach is a polynomial-time algorithm to
compute a connected, degree bounded multigraph of minimum cost.
We tackle this problem by generalizing a fundamental result of Kir\'aly, Lau
and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis
problem, and devise such an algorithm for general polymatroids, even allowing
element multiplicities.
Our result directly yields a -approximation to the metric
Many-visits TSP, as well as a -approximation for the problem of
scheduling classes of jobs with sequence-dependent setup times on a single
machine so as to minimize the makespan.Comment: arXiv admin note: text overlap with arXiv:1911.0989
Approximation Algorithms for Traveling Salesman Problems
The traveling salesman problem is the probably most famous problem in combinatorial optimization. Given a graph G and nonnegative edge costs, we want to find a closed walk in G that visits every vertex at least once and has minimum cost. We consider both the symmetric traveling salesman problem (TSP) where G is an undirected graph and the asymmetric traveling salesman problem (ATSP) where G is a directed graph. We also investigate the unit-weight special cases and the more general path versions, where we do not require the walk to be closed, but to start and end in prescribed vertices s and t. In this thesis we give improved approximation algorithms and better upper bounds on the integrality ratio of the classical linear programming relaxations for several of these traveling salesman problems. For this we use techniques arising from various parts of combinatorial optimization such as linear programming, network flows, ear-decompositions, matroids, and T-joins. Our results include a (22 + &epsilon)-approximation algorithm for ATSP (for any &epsilon > 0), the first constant upper bound on the integrality ratio for s-t-path ATSP, a new upper bound on the integrality ratio for s-t-path TSP, and a black-box reduction from s-t-path TSP to TSP
Traveling Salesman Problem
This book is a collection of current research in the application of evolutionary algorithms and other optimal algorithms to solving the TSP problem. It brings together researchers with applications in Artificial Immune Systems, Genetic Algorithms, Neural Networks and Differential Evolution Algorithm. Hybrid systems, like Fuzzy Maps, Chaotic Maps and Parallelized TSP are also presented. Most importantly, this book presents both theoretical as well as practical applications of TSP, which will be a vital tool for researchers and graduate entry students in the field of applied Mathematics, Computing Science and Engineering
The family traveling salesman problem
Consider a depot, a partition of the set of nodes into subsets, called families, and a cost matrix. The objective of the family traveling salesman problem (FTSP) is to find the minimum cost circuit that starts and ends at the depot and visits a given number of nodes per family. The FTSP was motivated by the order picking problem in warehouses where products of the same type are stored in different places and it is a recent problem. Nevertheless, the FTSP is an extension of well-known problems, such as the traveling salesman problem.
Since the benchmark instances available are in small number we developed a generator, which given a cost matrix creates an FTSP instance with the same cost matrix. We generated several test instances that are available in a site dedicated to the FTSP.
We propose several mixed integer linear programming models for the FTSP. Additionally, we establish a theoretical and a practical comparison between them. Some of the proposed models have exponentially many constraints, therefore we developed a branch-and-cut (B&C) algorithm to solve them. With the B&C algorithm we were able to obtain the optimal value of open benchmark instances and of the majority of the generated instances.
As the FTSP is an NP-hard problem we develop three distinct heuristic methods: a genetic algorithm, an iterated local search algorithm and a hybrid algorithm. With all of them we were able to improve the best upper bounds available in the literature for the benchmark instances that still have an unknown optimal value.
We created a new variant of the FTSP, called the restricted family traveling salesman problem (RFTSP), in which nodes from the same family must be visited consecutively. We apply to the RFTSP the methods proposed for the FTSP and develop a new formulation based on the interfamily and the intrafamily relationship
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