Non-Euclidean visual traveling salesman problem

Euclidean Traveling Salesman Problem

Since the computation of an exact solution is very expensive (under the assumption that P 6= NP ), we are interested in approximate solutions – at least for special cases. Particularly, we look for an Polynomial Time Approximation Scheme (PTAS), i.e., for any fixed error parameter ε > 0, the running time is bounded by a polynomial in n and the costs of the computed tour does not exceed (1+ε) ·O...

Lecture Notes: Euclidean Traveling Salesman Problem

Polynomial Time Approximation Schemes for the Euclidean Traveling Salesman Problem

In this report, we aim to understand the key ideas and major techniques used in the assigned paper "Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and Other Geometric Problems" by Sanjeev Arora. We also provide a review of related literature with an emphasis on the concurrent work by Joseph S. B. Mitchell. One presentation topic is selected from Arora’s paper and the oth...

The Traveling Salesman Problem under Squared Euclidean Distances

Let P be a set of points in R, and let α > 1 be a real number. We define the distance between two points p, q ∈ P as |pq|, where |pq| denotes the standard Euclidean distance between p and q. We denote the traveling salesman problem under this distance function by Tsp(d,α). We design a 5-approximation algorithm for Tsp(2,2) and generalize this result to obtain an approximation factor of 3 + √ 6 ...

The Euclidean Traveling Salesman Problem and a Space-filling Curve

We elucidate the relationship between space-llling curves and the Euclidean Traveling 2 Salesman Problem (TSP) by reference to a particular space-lling curve whose scaling behaviour is strongly related to the conjectured scaling behaviour of the optimal TSP tour. We suggest that space-lling curves can be used to generate testbed TSPs: sets of points which in the limit cover a planar surface and...