Approximating TSP on metrics with bounded global growth

The Traveling Salesman Problem (TSP) is a canonical NP-complete problem which is known to be MAXSNP hard even on Euclidean metrics (of high dimensions) [40]. In order to circumvent this hardness, researchers have been developing approximation schemes for low-dimensional metrics [4, 39] (under different notions of dimension). However, a feature of most current notions of metric dimension is that they are "local": the definitions require every local neighborhood to be well-behaved What if our metric looks a bit more realistic: it has a few "dense" regions, but is "well-behaved on the average"? We give a global notion of dimension that we call the correlation dimension (dimC), which generalizes the popular notion of doubling dimension: the class of metrics with dimC = O(1) not only contains all doubling metrics, but also some metrics containing uniform metrics of size n (but no larger). We first show that we can solve TSP (and other optimization problems) o...

• Type: Conference
• Year: 2008
• Where: SODA
• Authors: T.-H. Hubert Chan, Anupam Gupta
• Keywords: Algorithms, Canonical Np-complete Problem, Euclidean Metrics, Low-dimensional Metrics, SODA 2008