This paper studies a shuffled linear regression problem. As a variant of ordinary linear regression, it requires estimating not only the regression variable, but also permutational correspondences between the covariates and responses. While existing formulations require the underlying ground-truth correspondences to be an ideal bijection such that all pieces of data should match, such a requirement barely holds in real-world applications due to either missing data or outliers. In this work, we generalize the formulation of shuffled linear regression to a broader range of conditions where only a part of the data should correspond. To this end, the effective recovery condition and NP-hardness of the proposed formulation are also studied. Moreover, we present a simple yet effective algorithm for deriving the solution. Its global convergence property and convergence rate are also analyzed in detail. Distinct tasks validate the effectiveness of our proposed formulation and the solution method.