Dual Charging for Half-Integral TSP

We show that the max entropy algorithm is a randomized 1.49776 approximation for half-integral TSP, improving upon the previous known bound of 1.49993 from Karlin et al. This also improves upon the best-known approximation for half-integral TSP due to Gupta et al. Our improvement results from using the dual, instead of the primal, to analyze the expected cost of the matching. We believe this method of analysis could lead to a simpler proof that max entropy is a better-than-3/2 approximation in the general case. We also give a 1.4671 approximation for half integral LP solutions with no proper minimum cuts and an even number of vertices, improving upon the bound of Haddadan and Newman of 1.476. We then extend the analysis to the case when there are an odd number of vertices $n$ at the cost of an additional $O(1/n)$ factor.