On the Complexity of Four Polyhedral Set Containment Problems
On the Complexity of Four Polyhedral Set Containment Problems
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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.
What to read after On the Complexity of Four Polyhedral Set Containment Problems?
You can find similar books in the "Read Also" column, or choose other free books by Robert Michael Freund to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
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.
What to read after On the Complexity of Four Polyhedral Set Containment Problems?
You can find similar books in the "Read Also" column, or choose other free books by Robert Michael Freund to read onlineMoreLess
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