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Register a new userA historical note on the 3/2-approximation algorithm for the metric traveling salesman problem
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Added by
Ren\\'e van Bevern
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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One of the most fundamental results in combinatorial optimization is the polynomial-time 3/2-approximation algorithm for the metric traveling salesman problem. It was presented by Christofides in 1976 and is well known as the Christofides algorithm. Recently, some authors started calling it Christofides-Serdyukov algorithm, pointing out that it was published independently in the USSR in 1978. We provide some historic background on Serdyukovs findings and a translation of his article from Russian into English.
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We give a constant factor approximation algorithm for the asymmetric traveling salesman problem when the support graph of the solution of the Held-Karp linear programming relaxation has bounded orientable genus.
We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n / log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, Mk{a}dry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the 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.
We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $mathbb{R}^d$, with $dge 3$, are $mathrm{NP}$-hardness and an $O(log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $mathbb{R}^d$ is APX-hard for any $dge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1le k leq d-2$ unless $mathrm{P}=mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(log n)$ by showing that TSP with lines does not admit a $(2-epsilon)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(loglog n)}$.
With applications to many disciplines, the traveling salesman problem (TSP) is a classical computer science optimization problem with applications to industrial engineering, theoretical computer science, bioinformatics, and several other disciplines. In recent years, there have been a plethora of novel approaches for approximate solutions ranging from simplistic greedy to cooperative distributed algorithms derived from artificial intelligence. In this paper, we perform an evaluation and analysis of cornerstone algorithms for the Euclidean TSP. We evaluate greedy, 2-opt, and genetic algorithms. We use several datasets as input for the algorithms including a small dataset, a mediumsized dataset representing cities in the United States, and a synthetic dataset consisting of 200 cities to test algorithm scalability. We discover that the greedy and 2-opt algorithms efficiently calculate solutions for smaller datasets. Genetic algorithm has the best performance for optimality for medium to large datasets, but generally have longer runtime. Our implementations is public available.
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