W. Cook, J. Fonlupt, A. Schrijver, An integer analogue of Caratheodory's theorem, Journal of Combinatorial theory (B) 40 (1986) 63-70.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....cone as a non negative integer combination of at most 2n 2 irreducible vectors. and H(C) its minimal Hilbert basis. Every integral point in C can be written as the non negative integral combination of at most 2n 2 elements from H(C) Theorem 3. 10 improves a result of Cook, Fonlupt Schrijver [18] who showed that every integral vector in a pointed n dimensional cone is the non negative integral combination of at most 2n 1 vectors from the minimal Hilbert basis. From a result of Seb o [86] follows that in dimensions n = 2 and n = 3 every integral vector in a pointed n dimensional cone is ....

....from the minimal Hilbert basis. From a result of Seb o [86] follows that in dimensions n = 2 and n = 3 every integral vector in a pointed n dimensional cone is the non negative integral combination of at most n vectors from the Hilbert basis. This also holds for cones arising from perfect graphs [18] and a class of cones described in [46] However, in general at least n b1=6 nc elements of the Hilbert basis are needed to represent any integral vector in the cone. and H(C) its minimal Hilbert basis. In general at least n b1=6 nc elements from H(C) are needed in order to represent any ....

W. Cook, J. Fonlupt and A. Schrijver (1986), An integer analogue of Caratheodory's theorem, J. Comb. Theory (B) 40, 63 - 70.

....can write every integral vector in any pointed cone of dimension n as the non negative integer combination of at most 2n Gamma 2 vectors from the integral basis. This was shown by Sebo [Seb90] and gives currently the best bound in general; it improves the bound given by Cook, Fonlupt Schrijver [CFS86] by 1, yet is still quite far from what many researchers conjecture to be true, namely: every integral vector in a pointed cone is the non negative integer combination of at most n vectors of the integral basis. So far it has been verified only for dimensions less or equal than three, see [Seb90] ....

W. Cook, J. Fonlupt, and A. Schrijver, An integer analogue of Caratheodory's theorem, J. Comb. Theory (B) 40, 1986, 63--70.

....In fact we can write every integral vector in any pointed cone of dimension n as the non negative integer combination of at most 2n Gamma 2 vectors from the basis. This was shown by Sebo [S90] and gives currently the best bound in general; it improves the bound given by Cook, Fonlupt Schrijver [CFS86] by 1, yet is still quite far from what many researchers conjecture to be true, namely: every integral vector in a pointed cone is the non negative integer combination of at most n vectors of the Hilbert basis. We now prove that this integer Version of Caratheodory s Theorem holds for the ....

W. Cook, J. Fonlupt, and A. Schrijver, An integer analogue of Caratheodory's theorem, J. Comb. Theory (B) 40, 1986, 63--70.

.... 2 irreducible vectors. Theorem 3.10. 86] Let C be a pointed cone in R n and H(C) its minimal Hilbert basis. Every integral point in C can be written as the non negative integral combination of at most 2n 2 elements from H(C) Theorem 3. 10 improves a result of Cook, Fonlupt Schrijver [18] who showed that every integral vector in a pointed n dimensional cone is the non negative integral combination of at most 2n 1 vectors from the minimal Hilbert basis. From a result of Sebo [86] follows that in dimensions n = 2 and n = 3 every integral vector in a pointed n dimensional cone is ....

....from the minimal Hilbert basis. From a result of Sebo [86] follows that in dimensions n = 2 and n = 3 every integral vector in a pointed n dimensional cone is the non negative integral combination of at most n vectors from the Hilbert basis. This also holds for cones arising from perfect graphs [18] and a class of cones described in [46] However, in general at least n #1 6n# elements of the Hilbert basis are needed to represent any integral vector in the cone. Theorem 3.11. 12] Let C be a pointed cone in R n and H(C) its minimal Hilbert basis. In general at least n #1 6 n# elements ....

W. Cook, J. Fonlupt and A. Schrijver (1986), An integer analogue of Caratheodory's theorem, J. Comb. Theory (B) 40, 63 - 70.

.... of H(C) Let us remark that the question whether n elements of the Hilbert basis are sufficient to express any integral vector of the cone as a nonnegative integral combination (and thus having a nice counterpart to Carath eodory s theorem) has already been raised by Cook, Fonlupt Schrijver [CFS86]. Obviously, UHP) implies (UHC) and (UHC) implies (ICP) Sebo also verified (UHP) and thus all three conjectures) in dimensions n 3 [Seb90] An independent proof was given by Aguzzoli Mundici [AM94] in the context of desingularization of 3 dimensional toric varieties. However, in dimensions n ....

....= maxfCR(C) C ae R n an integral pointed polyhedral coneg be the maximal Carath eodory rank in dimension n. With this notation the (ICP) conjecture claims CR(C) n, or equivalent, h(n) n which holds in dimensions n 3. A first general upper bound on h(n) was given by Cook, Fonlupt Schrijver [CFS86]. They proved h(n) 2n Gamma 1 and they also verified the (ICP) conjecture for certain cones arising from perfect graphs. Another class of cones satisfying (ICP) is described in [HW97] The bound 2n Gamma 1 was improved by Sebo [Seb90] to h(n) 2n Gamma 2 which is currently the best known ....

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W. Cook, J. Fonlupt, and A. Schrijver, An integer analogue of Carath'eodory's theorem, J. Comb. Theory (B) 40 (1986), 63--70.

....(Z d ; C d ) had been studied before by Doignon [6] Also, some recent investigations motivated by integer programming were concerned with Helly properties of the integer lattice (Bell [1] Scarf [15] Schrijver [16, p. 234] and Caratheodory properties of it (Cook, Fonlupt, and Schrijver [4]) In contrast with the linear growth rate, as a function of the dimension, of the Radon number of the reals given above, we show in x2 an Omega Gamma d ) lower bound on the Radon number r(d) of the integer lattice. It is interesting to note that this lower bound is also in contrast with a ....

W. Cook, J. Fonlupt, and A. Schrijver, An integer analogue of Caratheodory's theorem, J. Combin. Theory Ser. B, 40 (1986), pp. 63-70.

....to the integer cone of S. Understandably, it is of special interest to know when equality holds in (1) Definition 1.1 A set of vectors S for which equality holds in (1) is called a Hilbert basis. This concept is closely related to total dual integrality, and has been studied by various authors [16, 6, 35, 36]. In our setting, the Hilbert basis problem is to determine classes of matroids and graphs for which C and M form Hilbert bases. This problem will be addressed in Sections 3 and 6, respectively. It must be emphasised that the cone and the lattice of S are worthy of independent study. For example, ....

W. Cook, J. Fonloupt and A. Schrijver, An integer analogue of Carath'eodory's Theorem, J. Combin. Theory Ser. B 40 (1986), 63-70.

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W. Cook, J. Fonlupt, A. Schrijver, An integer analogue of Caratheodory's theorem, Journal of Combinatorial theory (B) 40 (1986) 63-70.

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