Given vectors x and y in a separable complex Hilbert space $mathcal{H}$, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate compact interpolation problems for vectors in a tridiagonal algebra. We show the following : Let Alg$mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $mathcal{H}$ and let x = $(x_i)$ and y = $(y_i)$ be vectors in H. Then the following are equivalent: (1) There exists a compact operator A = $(a_{ij})$ in Alg$mathcal{L}$ such that Ax = y. (2) There is a sequence ${{alpha}_n}$ in $mathbb{C}$ such that ${{alpha}_n}$ converges to zero and for all k ${in}$ $mathbb{N}$, $y_1 = {alpha}_1x_1 + {alpha}_2x_2$ $y_{2k} = {alpha}_{4k-1}x_{2k}$ $y_{2k+1}={alpha}_{4k}x_{2k}+{alpha}_{4k+1}x_{2k+1}+{alpha}_{4k+2}+x_{2k+2}$.