One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to $\frac43$. For 40 years, the best known upper bound was 1.5, due to Wolsey (1980). Recently, Karlin, Klein, and Oveis Gharan (2022) showed that the max entropy algorithm for the TSP gives an improved bound of $1.5 - 10^{-36}$. In this paper, we show that the approximation ratio of the max entropy algorithm is at least 1.375, even for graphic TSP. Thus the max entropy algorithm does not appear to be the algorithm that will ultimately resolve the four-thirds conjecture in the affirmative, should that be possible.

翻译：组合优化领域最著名的猜想之一是四分之三猜想，即TSP子环LP松弛的整性间隙等于$\frac43$。四十年来，Wolsey（1980）证明的最佳上界为1.5。近期，Karlin、Klein和Oveis Gharan（2022）表明，TSP的最大熵算法可将上界改进至$1.5 - 10^{-36}$。本文证明，即使针对图论TSP，最大熵算法的近似比至少为1.375。因此，即使可能最终以肯定形式解决四分之三猜想，最大熵算法似乎也不是最终能实现这一目标的算法。