Matrix analysis
An optimal algorithm for finding minimal enclosing triangles
Journal of Algorithms
Computational complexity of inner and outer j-radii of polytopes in finite-dimensional normed spaces
Mathematical Programming: Series A and B
On the complexity of some basic problems in computational convexity: I.: containment problems
Discrete Mathematics - Special issue: trends in discrete mathematics
Algorithms for minimum volume enclosing simplex in R3
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Largest parallelotopes contained in simplices
Discrete Mathematics
Oracle-polynomial-time approximation of largest simplices in convex bodies
Discrete Mathematics - Selected papers in honor of Ludwig Danzer
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Introduction to the Theory of Complexity and Approximation Algorithms
Lectures on Proof Verification and Approximation Algorithms. (the book grow out of a Dagstuhl Seminar, April 21-25, 1997)
Information Processing Letters
Efficient simplex computation for fixture layout design
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Efficient simplex computation for fixture layout design
Computer-Aided Design
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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 that vol2(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.