We present a new combinatorial algorithm for the optimal general topological simplification of scalar fields on surfaces. Given a piecewise linear (PL) scalar field f , our algorithm generates a simplified PL field g that provably admits critical points only from a constrained subset of the singularities of f while minimizing the distance || f − g||∞ for data-fitting purpose. In contrast to previous algorithms, our approach is oblivious to the strategy used for selecting features of interest and allows critical points to be removed arbitrarily and additionally minimizes the distance || f − g||∞ in the PL setting. Experiments show the generality of the algorithm as well as its time-efficiency, and demonstrate in practice the minimization of || f − g||∞ .