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The complexity of Philip Wolfe's method for the minimum Euclidean-norm pointproblem over a convex polytope has remained unknown since he proposed themethod in 1974. The method is important because it is used as a subroutine forone of the most practical algorithms for submodular function minimization. Wepresent the first example that Wolfe's method takes exponential time.Additionally, we improve previous results to show that linear programmingreduces in strongly-polynomial time to the minimum norm point problem over asimplex.

The Minimum Euclidean-Norm Point on a Convex Polytope: Wolfe's Combinatorial Algorithm is Exponential