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I want to solve a linear program using the simplex method, and I know that all my inequalities will pass through the origin (therefore, either my initial solution of (0, ... , 0) is optimal, or the program is unbounded; I'm only really interested in distinguishing between these two cases).

So here's how I think the algorithm would work: Step one is to pick a pivot variable. Step two - and here's where I think things break down - is to find the equation with the smallest value of the constant to the coefficient of the pivot variable. But since my inequalities all pass through the origin, the constant is 0 every time, so all equations are an equally valid choice for the pivot. This reduces the Simplex Method to a brute-force search of the set of basic variables, which would make it run in above-polynomial time.

Is this correct? Or am I missing some feature of the Simplex Method that handles this case?

Thanks in advance!

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You might find it interesting to do a google search on Klee-Minty cube. – Patricia Hersh Oct 9 '12 at 22:26
up vote 2 down vote accepted

As far as I can recall, the Simplex Algorithm is not running in polynomial time, although when randomly perturbing the input, it runs on average in polynomial time (this is also studied under the name "Smoothed Analysis of Algorithms", see for example the article by Spielman and Teng, Journal of the ACM, Vol. 51, No. 3, May 2004, pp. 385–463

There are however algorithms for linear programming which run in polynomial time, known as "inner-point" algorithms, because they traverse the interior of the simplex.

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I looked into it more - you're completely right, and now I'm trying to remember why I was so sure Simplex was strongly polynomial in the first place. I may have found one of the cases on which it's exponential. I found a source that says there are no known strongly-polynomial linear programming algorithms, and the existence of such an algorithm is a major open question. Thanks for your help! – user21816 Apr 24 '12 at 21:45

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