P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. II. Volume and mixed volumes, in: Polytopes: abstract, convex and computational (Scarborough, ON,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....polarization formula [8, A.8) p. 353] any mixed volume of rational polytopes is rational. This shows that kwkb P (w=kwk) is rational. Clearly any representation of P itself cannot be part of the input. To handle such situations, the notion of an oracle has been developed; see, for example, [12] and [15, p. 26] We assume that the unknown rational convex polytope P in R is well bounded, that is, rB P RB for some rational numbers 0 r R. The oracle we require has the following input and output: Oracle input: A rational vector w in R . Oracle output: The rational kwkb P ....

....be the sum of hri, hRi, and the sizes of the vertices of P . Here we are assuming the oracle has access to P in its V representation. This is not important, however, since a V representation can be converted to an H representation, and vice versa, in polynomial time when the dimension is xed; see [12]. It is a consequence of [13, 4.2.1] that when n is xed, V (P; n 1; o; w] and hence kwkb P (w=kwk) can be computed in a time bounded by a polynomial in hP i and hwi. Note that with our formulation of the oracle, it does not conform to the general assumption about oracles in [15, p. 26] that ....

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. I. Containment problems, Discrete Math. 13 (1994), 129-174.

....#R#, and the sizes of the vertices of P . Here we are assuming the oracle has access to P in its V representation. This is not important, however, since a can be converted to an and vice versa, in polynomial time when the dimension is fixed; see [12] It is a consequence of 4.2. 1 of [13] that when n is fixed, V (P, n 1; o,w] and hence P (w #w#) can be computed in a time bounded by a polynomial in #w#. Note that with our formulation of the oracle, it does not conform to the general assumption about oracles on p. 26 of [15] that the size of the oracle output is bounded ....

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. II. Volume and mixed volumes, in: Polytopes: Abstract, Convex and Computational (Scarborough, ON,

....polarization formula [8, A.8) p. 353] any mixedvolume of rational polytopes is rational. This shows that P (w #w#) is rational. Clearly any representation of P itself cannot be part of the input. To handle such situations, the notion of an oracle has been developed; see, for example, [12] and p. 26 of [15] We assume that the unknown rational convex polytope P in R n is well bounded, that is, rB RB for some rational numbers 0 r R. The oracle we require has the following input and output: Oracle input: A rational vector w in R n . Oracle output: The rational P ....

....P is defined to be the sum of #r#, #R#, and the sizes of the vertices of P . Here we are assuming the oracle has access to P in its V representation. This is not important, however, since a can be converted to an and vice versa, in polynomial time when the dimension is fixed; see [12]. It is a consequence of 4.2.1 of [13] that when n is fixed, V (P, n 1; o,w] and hence P (w #w#) can be computed in a time bounded by a polynomial in #w#. Note that with our formulation of the oracle, it does not conform to the general assumption about oracles on p. 26 of [15] that ....

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. I. Containment problems, Discrete Math. 13 (1994), 129--174.

....polarization for mula [8, A.8) p. 353] any mixed volume of rational polytopes is rational. This shows that Ilwllb(w llwll) is rational. Clearly any representation of P itself cannot be part of the input. To handle such situations, the notion of an oracle has been developed; see, for example, [12] and [15, p. 26] We assume that the unknown rational convex polytope P in 11 is well bounded, that is, rB C P C RB for some rational numbers 0 r R. The oracle we require has the following input and output: Oracle input: A rational vector w in 11 . Oracle output: The rational ....

....the sum of (r , R , and the sizes of the vertices of P. Here we are assuming the oracle has access to P in its F representation. This is not important, however, since a F representation can be converted to an 7 representation, and vice versa, in polynomial time when the dimension is fixed; see [12]. It is a consequence of [13, 4.2.1] that when n is fixed, V(P, n 1; o, w] and hence IIwllb(w llwll) can be computed in a time bounded by a polynomial in (P and Note that with our formulation of the oracle, it does not conform to the general assumption about oracles in [15, p. 26] that the ....

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. I. Containment problems, Discrete Math. 13 (1994), 129-174.

....so called polarization formula [8, A.8) any mixed volume of rational polytopes is rational. This shows that kwkb P (w=kwk) is rational. Clearly any representation of P itself cannot be part of the input. To handle such situations, the notion of an oracle has been developed; see, for example, [12] and the references given there. We assume that the unknown rational convex polytope P in R n is well bounded, that is, rB ae P ae RB for some rational numbers 0 r R. The oracle we require has the following input and output: Oracle input: A rational vector w in R n . Oracle output: The ....

....sizes of r, R, and the components of the vertices of P . Here we are assuming the oracle has access to P in its V representation. This is not important, however, since a V representation can be converted to an H representation, and vice versa, in polynomial time when the dimension is fixed; see [12]. The size of the input to the oracle is the size of P plus the size of w. It is a consequence of [13, 4.2.1] that V (P; n Gamma 1; o; w] and hence kwkb P (w=kwk) can be computed in polynomial time when the dimension is fixed. With this background, we can now describe the algorithmic problem ....

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. I. Containment problems, Discrete Comput. Geom. 13 (1994), 129--174.

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P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. II. Volume and mixed volumes, in: Polytopes: abstract, convex and computational (Scarborough, ON,

No context found.

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. II. Volume and mixed volumes, in: Polytopes: abstract, convex and computational (Scarborough, ON,

No context found.

P. Gritzmann and V. Klee, On the complexity of some basic problems in computational geometry. II. Volume and mixed volumes, in: Polytopes: abstract, convex and computational (Scarborough, ON,

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