On the Complexity of Four Polyhedral Set Containment Problems
On the Complexity of Four Polyhedral Set Containment Problems
Robert Michael Freund
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If X is an H-cell, this is the usual linear program, whose solution time, while polynomial, is by no means negligible. However, if X is represented as a W-cell, the linear programming problem becomes trivial. As another example, consider the problem of testing if xeX for a given x, where X is a polyhedron. If X is an H-cell, the problem is trivial, whereas if X is a W-cell, the problem reduces to solving a linear program. This note discusses the complexity of two types of problems. The first pr...oblem is the set containment problem (SCP), that of determining if XcY, where X (resp. Y) is a cell, defined to be either a polyhedron (an H-cell or a W-cell), t 2 or a closed solid ball of the form {x: (x-c) (x-c) (2max(A) ^^ "^^ nl ) . PROOF. Let z*= d(P(A), {-1, 1}'^), and z*(y)= d(P(A), {y}) ; then z*=min (z* (y) :ye { -1, 1} ), and z*(y)= minimum z, subject to z + (x. -y. ) >^0 j = l, .. , n t^j-yj) ->-0 j = i, n xeP(A) We now claim that z* (y)> (2inax(A) (nl)) for any Ye{-l, l} . To see this, let X* be a point in P(A) of minimum distance to y, and without loss of generality we may assume that x* is an extreme point of the feasible region of the above linear program.
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