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TSP instances that require exponentially many subtour elimination constraints
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We construct pathological TSP instances that require exponentially many subtour elimination constraints (SECs) in the sense that, if a smaller collection of SECs is imposed, then the Dantzig-Fulkerson-Johnson model returns a disconnected solution costing less than an optimal tour. For asymmetric instances on $n$ vertices, we prove a growth rate of \(\Omega(\phi^n)\), where \(\phi\approx 1.618\) is the golden ratio. A node-splitting reduction gives metric instances with \(\Omega(\phi^{n/2})\) growth. We also give instances in the plane under the Euclidean, Rectilinear, and Maximum metrics that require exponentially many SECs. Concorde struggles on them, with solve times growing exponentially despite tiny integrality gaps.