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On the complexity of four polyhedral set containment problems. (English) Zbl 0581.90060
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
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##### References:
  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  F.R. Gantmacher,Matrix theory, vol. 1 (Chelsea, New York, 1959). · Zbl 0085.01001  M.R. Garey and D.S. Johnson,Computers and intractability (W.H. Freeman, San Francisco, 1979).  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.
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