We show that the Simplex Method, the Network Simplex Method—both with Dantzig’s original pivot rule—and the Successive Shortest Path Algorithm are NP-mighty. That is, each of these algorithms can be used to solve, with polynomial overhead, any problem in NP implicitly during the algorithm’s execution. This result casts a more favorable light on these algorithms’ exponential worst-case running times. Furthermore, as a consequence of our approach, we obtain several novel hardness results. For example, for a given input to the Simplex Algorithm, deciding whether a given variable ever enters the basis during the algorithm’s execution and determining the number of iterations needed are both NP-hard problems. Finally, we close a long-standing open problem in the area of network flows over time by showing that earliest arrival flows are NP-hard to obtain.