Polynomial-time approximation of largest simplices in V-polytopes

Abstract

This paper considers the problem of computing the squared volume of a largest j-dimensional simplex in an arbitrary d-dimensional polytope P given by its vertices (a “V-polytope”), for arbitrary integers j and d with 1⩽j⩽d. The problem was shown by Gritzmann, Klee and Larman to be NP-hard. This paper examines the possible accuracy of deterministic polynomial-time approximation algorithms for the problem. On the negative side, it is shown that unless P=NP, no such algorithm can approximately solve the problem within a factor of less than 1.09. It is also shown that the NP-hardness and inapproximability continue to hold when the polytope P is restricted to be an affine crosspolytope.On the positive side, a simple deterministic polynomial-time approximation algorithm for the problem is described. The algorithm takes as input integers j and d with 1⩽j⩽d and a V-polytope P of dimension d. It returns a j-simplex S⊂P such thatvol2(T)vol2(S)⩽A(Bj)j,where T is any largest j-simplex in P, and A and B are positive constants independent of j,d,P.