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We show that the shadow vertex simplex algorithm can be used to solve linearprograms in strongly polynomial time with respect to the number $n$ ofvariables, the number $m$ of constraints, and $1/\delta$, where $\delta$ is aparameter that measures the flatness of the vertices of the polyhedron. Thisextends our recent result that the shadow vertex algorithm finds paths ofpolynomial length (w.r.t. $n$, $m$, and $1/\delta$) between two given verticesof a polyhedron.  Our result also complements a recent result due to Eisenbrand and Vempala whohave shown that a certain version of the random edge pivot rule solves linearprograms with a running time that is strongly polynomial in the number ofvariables $n$ and $1/\delta$, but independent of the number $m$ of constraints.Even though the running time of our algorithm depends on $m$, it issignificantly faster for the important special case of totally unimodularlinear programs, for which $1/\delta\le n$ and which have only $O(n^2)$constraints.

Solving Totally Unimodular LPs with the Shadow Vertex Algorithm