On the complexity of four polyhedral set containment problems

A nonempty closed convex polyhedron X can be represented either as X = {x: Ax < b}, where (A, b) are given, in which case X is called an H-cell, or in the form X= {x: x = UA + V , A1, A 0, z 0}, where (U, V) are given, in which case X is called a W-cell. This note discusses the computational complexity of certain set containment problems. The problems of determining if X e Y, where (i) X is an H-cell and Y is a closed solid ball, (ii) X is an H-cell and Y is a W-cell, or (iii) X is a closed solid ball and Y is a W-cell, are all shown to be NP-complete, essentially verifying a conjecture of Eaves and Freund. Furthermore, the problem of determining whether there exists an integer poiAt in a W-cell is shown to be NP-complete, demonstrating that regardless of the representation of X as an H-cell or W-cell, this integer containment problem is NP-complete.