On the complexity of four polyhedral set containment problems. (English) Zbl 0581.90060

Freund, Robert M.; Orlin, James B.

Math. Program. 33, 139-145 (1985).

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 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.

MSC:

90C10	Integer programming
52Bxx	Polytopes and polyhedra
68Q25	Analysis of algorithms and problem complexity

Keywords:

H-cell; W-cell; computational complexity; set containment problems; NP- complete

References:

[1]	B.C. Eaves and R.M. Freund, ”Optimal scaling of balls and polyhedra”,Mathematical Programming 23 (1982) 138–147. · Zbl 0479.90064 · doi:10.1007/BF01583784
[2]	F.R. Gantmacher,Matrix theory, vol. 1 (Chelsea, New York, 1959). · Zbl 0085.01001
[3]	M.R. Garey and D.S. Johnson,Computers and intractability (W.H. Freeman, San Francisco, 1979).
[4]	R.M. Karp, ”Reducibility among combinatorial problems”, in: R.E. Miller and J.W. Thatcher, eds.,Complexity of computer computations (Plenum Press, New York, 1972) pp. 85–103.