Cell-free massive MIMO systems are currently being considered as potential enablers of future (6G) technologies for wireless communications. By combining distributed processing and massive MIMO, they are expected to deliver improved user coverage and efficiency. A possible source of performance degradation in such systems is pilot contamination, which contributes to causing interference during uplink training and affects channel estimation negatively. Contamination occurs when the same pilot sequence is assigned to more than one user. This is in general inevitable, as the number of mutually orthogonal pilot sequences corresponds to only a fraction of the coherence interval. We introduce a new algorithm for pilot assignment and analyze its performance both from a theoretical perspective and in computational experiments. We show that it has an approximation ratio close to 1 for a plausibly large number of orthogonal pilot sequences, as well as low computational complexity under massive parallelism. We also show that, on average, it outperforms other methods in terms of per-user SINR and throughput on the uplink.