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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 $n^{O(loglog n)}$.
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
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.
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 spanni
A new characterisation of Hamiltonian graphs using f-cutset matrix is proposed. A new exact polynomial time algorithm for the travelling salesman problem (TSP) based on this new characterisation is developed. We then define so called ordered weighted