Let C be a compact set in ℝ² and let S be a set of n points in ℝ². We consider the problem of computing a translate of C that contains the maximum number, κ*, of points of S. It is known that this problem can be solved in a time that is roughly quadratic in n. We show how random-sampling and bucketing techniques can be used to develop a near-linear-time Monte Carlo algorithm that computes a placement of C containing at least (1 - ɛ)κ*. points of S, for given ɛ > 0, with high probability. We also present a deterministic algorithm that solves the ?-approximate version of the optimal-placement problem and runs in O((n ^1+δ+ n/ɛ) logm) time, for arbitrary constant δ > 0, if C is a convex m-gon.