Exponential Time Approximation Scheme for TSP

Zhixiang Chen, Qilong Feng, Bin Fu, Mugang Lin, Jianxin Wang

Abstract: In this paper, we develop an exponential time approximation scheme for the traveling salesman problem (TSP) on undirected graphs. If the weight of each edge is a nonnegative real number, then there is an algorithm to give an \((1+\epsilon )\) approximation for the TSP problem in \(O({1\over \epsilon }\cdot 1.66^n)\) and a polynomial space. It is in contrast to Golovnen’s approximation scheme for TSP on directed graphs with \(\mathrm{O}({1\over \epsilon }\cdot 2^n)\) time. We also show that there is no \(2^{o(n)}\) time constant factor approximation for the TSP problem under Exponential Time Hypothesis in complexity theory.