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I'm reading Efficient Shortest Path Simplex Algorithms by Donald Goldfarb, Jianxiu Hao and Shen-Roan Kai who considered "the specialization of the primal simplex algorithm to the problem of finding a tree of directed shortest paths from a given node to all other nodes in a network of n nodes or finding a directed cycle of negative length. Two efficient variants of this shortest path simplex algorithm are analyzed and shown to require at most $(n − 1)(n − 2)/2$ pivots and $O(n^3)$ time."

I'm trying to find motivation for this article and wonder isn't Bellman-Ford algorithm good enough? It works in $O(nm)$ time which and good for the type of graph which the problem above algorithm deals with.

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up vote 16 down vote accepted

A major open problem in mathematical programming is designing a strongly polynomial time linear programming algorithm. A related problem is whether any variant of the simplex algorithm runs in strongly polynomial time. It makes sense to first prove strong polynomial time bounds for variants of simplex applied to problems for which we already know strong polynomial time algorithms exist.

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