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In discrete choice experiments, the information matrix depends on the model parameters. Therefore designing optimally informative experiments for arbitrary initial parameters often yields highly nonlinear optimization problems and makes optimal design infeasible. To overcome such challenges, we connect design theory for discrete choice experiments with Laplacian matrices of undirected graphs, resulting in complexity reduction and feasibility of optimal design. We rewrite the $D$-optimality criterion in terms of Laplacians via Kirchhoff's matrix tree theorem, and show that its dual has a simple description via the Cayley-Menger determinant of the Farris transform of the Laplacian matrix. This results in a drastic reduction of complexity and allows us to implement a gradient descent algorithm to find locally $D$-optimal designs. For the subclass of Bradley-Terry paired comparison models, we find a direct link to maximum likelihood estimation for Laplacian-constrained Gaussian graphical models. Finally, we study the performance of our algorithm and demonstrate its application to real and simulated data.