In defense of the Simplex Algorithm's worst-case behavior.
Yann Disser, Martin Skutella · 2013
In the early 1970s, by work of Klee and Minty (1972) and Zadeh (1973), the Simplex Method, the Network Simplex Method, and the Successive Shortest Path Algorithm have been proved guilty of exponential worst-case behavior (for certain pivot rules). Since then, the common perception is that these algorithms can be fooled into investing senseless effort by ‘bad instances’ such as, e. g., Klee-Minty cubes. This paper promotes a more favorable stance towards the algorithms’ worstcase behavior. We argue that the exponential worst-case performance is not necessarily a senseless waste of time, but may rather be due to the algorithms performing meaningful operations and solving difficult problems on their way. Given one of the above algorithms as a black box, we show that using this black box, with polynomial overhead and a limited interface, we can solve any problem in NP. This also allows us to derive NP-hardness results for some related problems.