The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with intercept $-κ$. We analyze the performance of the canonical discrepancy minimization algorithms of Lovett-Meka and Rothvoss/Eldan-Singh for the asymmetric binary perceptron problem. We obtain new algorithmic results in the $κ= 0$ case and in the large-$|κ|$ case. In the $κ\to-\infty$ case, we additionally characterize the storage capacity and complement our algorithmic results with an almost-matching overlap-gap lower bound.