NP -Hardness of Largest Contained and Smallest Containing Simplices for V- and H-Polytopes

The problem of finding a d -simplex of maximum volume in an arbitrary d -dimensional V -polytope, for arbitrary d , was shown by Gritzmann et al. [GKL] in 1995 to be NP-hard. They conjectured that the corresponding problem for H -polytopes is also NP-hard. This paper presents a unified way of proving the NP-hardness of both these problems. The approach also yields NP-hardness proofs for the problems of finding d -simplices of minimum volume containing d -dimensional V - or H -polytopes. The polytopes that play the key role in the hardness proofs are truncations of simplices. A construction is presented which associates a truncated simplex to a given directed graph, and the hardness results follow from the hardness of detecting whether a directed graph has a partition into directed triangles.

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