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Register a new userTravelling salesman paths on Demidenko matrices
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Added by
Eranda Cela
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s, visit every city exactly once, and finally end his trip at another given city t. In this paper we identify a new polynomially solvable case of the Path-TSP where the distance matrix of the cities is a so-called Demidenko matrix. We identify a number of crucial combinatorial properties of the optimal solution, and we design a dynamic program with time complexity $O(n^6)$.
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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.
The Double Travelling Salesman Problem with Multiple Stacks, DTSPMS, deals with the collect and delivery of n commodities in two distinct cities, where the pickup and the delivery tours are related by LIFO constraints. During the pickup tour, commodities are loaded into a container of k rows, or stacks, with capacity c. This paper focuses on computational aspects of the DTSPMS, which is NP-hard. We first review the complexity of two critical subproblems: deciding whether a given pair of pickup and delivery tours is feasible and, given a loading plan, finding an optimal pair of pickup and delivery tours, are both polynomial under some conditions on k and c. We then prove a (3k)/2 standard approximation for the MinMetrickDTSPMS, where k is a universal constant, and other approximation results for vario
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 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 $n^{O(loglog n)}$.
Frank-Wolfe methods are popular for optimization over a polytope. One of the reasons is because they do not need projection onto the polytope but only linear optimization over it. To understand its complexity, Lacoste-Julien and Jaggi introduced a condition number for polytopes and showed linear convergence for several variations of the method. The actual running time can still be exponential in the worst case (when the condition number is exponential). We study the smoothed complexity of the condition number, namely the condition number of small random perturbations of the input polytope and show that it is polynomial for any simplex and exponential for general polytopes. Our results also apply to other condition measures of polytopes that have been proposed for the analysis of Frank-Wolfe methods: vertex-facet distance (Beck and Shtern) and facial distance (Pe~na and Rodriguez). Our argument for polytopes is a refinement of an argument that we develop to study the conditioning of random matrices. The basic argument shows that for $c>1$ a $d$-by-$n$ random Gaussian matrix with $n geq cd$ has a $d$-by-$d$ submatrix with minimum singular value that is exponentially small with high probability. This has consequences on results about the robust uniqueness of tensor decompositions.
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