Before revision:  We further generalize this to exhibit a O(1{\tau})-approximation } algorithm in O(m^{ \alpha\lfloor{\sqrt{12}}}\rfloor k}n\log k) time for a given grid width 1 \leq \tau \leq 2, where \alpha=O(\tau^2).
After revision:  We further generalize this to exhibit a O(1{\tau})-approximation } algorithm in O(m^{ \alpha\lfloor\frac{4\pi+8\tau+\tau^2{\sqrt{12}}}\rfloor k}n\log k) time for a given grid width 1 \leq \tau \leq 2, where \alpha=O(\tau^2).