  In this paper, we study the restriction problem for one class of
hypersurfaces with vanishing curvature in $\mathbb{R}^n$ with $n$ being odd. We
obtain an $L^2-L^p$ restriction estimate, which is optimal except at the
endpoint. Furthermore, we establish an $L^s-L^p$ restriction estimate for these
hypersurfaces, which is achieved by improving the known $L^{\infty}$
restriction estimate for hypersurfaces with $\frac{n-1}{2}$ positive principal
curvatures and $\frac{n-1}{2}$ negative principal curvatures.
