an:03930723
Zbl 0581.90060
Freund, Robert M.; Orlin, James B.
On the complexity of four polyhedral set containment problems
EN
Math. Program. 33, 139-145 (1985).
00150425
1985
j
90C10 52Bxx 68Q25
H-cell; W-cell; computational complexity; set containment problems; NP- complete
A nonempty closed convex polyhedron X can be represented either as \(X=\{x:\) Ax\(\leq b\}\), where (A,b) are given, in which case X is called an H-cell, or in the form \(X=\{x:\) \(x=U\lambda +V\mu\), \(\sum \lambda_ j=1\), \(\lambda\geq 0\), \(\mu\geq 0\}\), 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 \(X\not\subseteq Y\), 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 \textit{B. C. Eaves} and the first author [ibid. 23, 138-147 (1982; Zbl 0479.90064)]. Furthermore, the problem of determining whether there exists an integer point 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.
Zbl 0479.90064