We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a @math -H "older continuous map @math , with @math . Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in @math or @math in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as H "older curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves @math . We then find conditions in terms of tube approximations that ensure the approximating curves converge to a H "older curve. As an application, we provide sufficient conditions that guarantee fractional rectifiability of pointwise doubling measures in @math .
There is considerable interest in finding higher-dimensional analogues of the Analyst's Traveling Salesman theorem, for example finding a characterization of subsets of Lipschitz images of @math . This problem is still open, but some positive steps were recently taken by Azzam and Schul @cite_25 for . Also see @cite_37 .
@cite_37 @cite_25
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The Travelling Salesman Problem is one the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3 2, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only 4 3. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis , and then by Momke and Svensson. In this paper, we provide an improved analysis for the approach introduced by Momke and Svensson yielding a bound of 13 9 on the approximation factor, as well as a bound of 19 12+epsilon for any epsilon>0 for a more general Travelling Salesman Path Problem in graphic metrics. In this paper, new lower bounds for the asymmetric travelling salesman problem are presented, based on spanning arborescences. The new bounds are combined in an additive procedure whose theoretical performance is compared with that of the Balas and Christofides procedure (1981). Both procedures have been imbedded in a simple branch and bound algorithm and experimentally evaluated on hard test problems. We show that the traveling salesman problem with triangle inequality cannot be approximated with a ratio better than @math when the edge lengths are allowed to be asymmetric and @math when the edge lengths are symmetric, unless P=NP. The best previous lower bounds were @math and @math respectively. The reduction is from Hastad’s maximum satisfiability of linear equations modulo 2, and is nonconstructive. The traveling salesman problem is one of the most famous combinatorial problems. We identify a natural parameter for the two-dimensional Euclidean traveling salesman problem. We show that for random problems there is a rapid transition between soluble and insoluble instances of the decision problem at a critical value of this parameter. Hard instances of the traveling salesman problem are associated with this transition. Similar results are seen both with randomly generated problems and benchmark problems using geographical data. Surprisingly, finite-size scaling methods developed in statistical mechanics describe the behaviour around the critical value in random problems. Such phase transition phenomena appear to be ubiquitous. Indeed, we have yet to find an NP-complete problem which lacks a similar phase transition.