Core Concepts
The Analyst's Traveling Salesman algorithm has a polynomial time complexity of O(n^3), where n is the number of points in the input set.
Abstract
The content discusses the time complexity of the Analyst's Traveling Salesman algorithm, which is a generalization of the classic Traveling Salesman Problem (TSP). The key points are:
The Traveling Salesman Problem is a well-known NP-hard problem, meaning it is computationally difficult to solve optimally. Researchers have explored "nearly-optimal" algorithms that produce a path that is close in length to the optimal one.
The Analyst's Traveling Salesman Problem (ATSP) asks to find a curve of finite length that contains a given (finite or infinite) set of points in R^N. The classification of sets for which the ATSP can be solved was done by Jones and Okikiolu.
The authors show that the ATSP algorithm, in the case of a finite set of n points, has a polynomial time complexity of O(n^3). This is an improvement over the previous algorithms, which had a time complexity that depended on the dimension N.
The authors also discuss the sharpness of the exponent 3 in the time complexity, showing that it cannot be lowered.
The ATSP algorithm constructs a series of connected graphs Gk that approximate the input set V, and then uses these graphs to compute a tour that contains all the points in V.