Abstract: A classic result from Christofides in the 70s tells us that a fast algorithm for the traveling salesperson problem (TSP) exists which returns a solution at most 3/2 times worse than the optimal. Since then, however, no better approximation algorithm has been found. In this talk, I will give an overview of research towards the goal of beating 3/2 and will present the first sub-3/2 approximation algorithm for the special case of "half integral" TSP instances. These instances have received significant attention in part due to a conjecture from Schalekamp, Williamson and van Zuylen that they attain the integrality gap of the subtour polytope. If this conjecture is true, our work shows that the integrality gap of the polytope is bounded away from 3/2, giving hope for an improved approximation for the general case. This presentation is of joint work with Anna Karlin and Shayan Oveis Gharan.
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