We can do convolution in $O(n\log n)$ for plus/multiply polynomials with FFT. However, the approach doesn't seem very generalisable to rings in general. Has there been any progress over the naive $O(n^2)$ convolution for the max/plus ring?
I should note that one can transform soft-max/plus into plus/product by doing exponentiation. Here $\text{soft-max}(x,y)=\log(e^x+e^y) = \max(x,y) + \log(1+e^{\min(x,y)-\max(x,y)})$.