P. Conti and C. Traverso. Buchberger algorithm and integer programming. In AAECC-9, volume 539 of LNCS, pages 130--139. Springer, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A and a term order induced by c yields a test set for the familiy of integer programs IP (b) minfc associated with a xed matrix A 2 Z and varying b 2 Z . The connection between test sets for integer programming and Gr obner bases of toric ideals was rst established by Conti Traverso [15]. This algebraic view of test sets is important from an algorithmic point of view. Reduced Gr obner bases of toric ideals can be computed by the Buchberger algorithm [13] Reinterpreting the steps of this algorithm as operations on lattice vectors yields a combinatorial algorithm for computing ....

....and z are vertices of the polyhedron convfx 2 Z Ax = Az g. Moreover, the line with endpoints z and z is an edge of the polyhedron convfx 2 Z g. ii) Let z and z be two adjacent vertices of convfx 2 Z g, then (z ) gcd(z ) 2 G. We mentioned that Conti Traverso [15] established the connection between test sets of integer programs and Gr obner bases of toric ideals. The latter objects can be computed by the Buchberger algorithm [13] We discuss below a combinatorial variant of this procedure that allows us to determine a superset of the Graver test set and ....

P. Conti and C. Traverso (1991), Buchberger algorithm and integer programming, Proceedings AAECC9 (New Orleans), Springer LNCS 539, 130 - 139.

....d 3 3 6. The second example problem is of type (B) which can be seen by taking the vertex edge incidence matrix of the complete graph in m nodes and the vector w equals wi,j. In this case ( and d: There are several methods to attack such problems [43] but a new approach, presented in [12] and extended in [51] is to try to build a directed graph with the vertices of the fiber 71 1 (b) Note that for each finite subset T of the integer kernel(rA) F , we can define a graph Gb, on the fiber rl(b) Its vertices are the elements of rl(b) and there is a edge ( if and only if ....

P. Conti and C. Traverso, Buchberger algorithm and integer programming, Proceedings AAECC-9 (New Orleans), Springer Verlag LNCS, 539, 1991, 130-139.

....to perform column substitutions. In each of the substitution steps of the algorithm, we need to compute all the irreducible solutions z # F N with z j 0 (for some variable index j) This can, at least in principle, be accomplished with a combinatorial Buchberger type algorithm, see the papers [CT91, Tho95,UWZ97] and the survey [Wei98] We omit all the details of such an algorithm here because this approach is computationally intractable in general, even for very small instances. In fact, also the finiteness proof (Theorem 4.2) suggests that the substitution steps of the algorithm are too strong : At ....

P. Conti and C. Traverso, Buchberger algorithm and integer programming, Proceedings of the AAECC-9, Lecture Notes in Computer Science 539, Springer, Berlin, 1991, pp. 130--139. A Primal All-Integer Algorithm Based on Irreducible Solutions 41

....Stanley s proofs of the upper bound conjecture [21] and the g theorem [22] using the theory of Cohen Macaulay rings and toric varieties. In algebraic geometry the theory of toric varieties provides an important class of examples, with connections to still more areas, such as integer programming [4], 25] and mirror symmetry in mathematical physics [3] 6] A less obvious connection between the areas is through the theory of Hilbert schemes. This connection is most spectacularly illustrated in Mark Haiman s recent (2000) proof of the n conjecture of algebraic combinatorics [10] using the ....

Pasqualina Conti and Carlo Traverso. Buchberger algorithm and integer programming. In Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA,

....to a finite one. To this end we apply certain level sets, which are constructed by solving the continuous relaxation of (1) i.e. the stochastic program where the integer requirements in the second stage are dropped) Applying Grobner bases to solve integer linear programs was first proposed in [7] (see also [31] It yields additional information at a possibly high computational cost, so that usually it is an ine#cient method to solve the problem for a fixed righthand side. However, the additional information turns out to be highly beneficial when an integer linear program needs to be ....

....after introducing slack variables, the problem (with properly adjusted q, W and y) is in equality form (P) min qy : Wy= s, y # ZZ m , with m = m p. For problems of this type, recently a solution technique has been developed basedonGrobner basis methods from computational algebra [7]. For a brief exposition of the method we introduce the problem (IP) min cy : Ay = b, y # ZZ n , where A is an integral d n matrix and c, b are integral vectors of dimensions n and d, respectively. To keep the exposition simple we assume here that all entries in c, A and b are ....

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P. Conti and C. Traverso. Buchberger algorithm and integer programming. In Proceedings AAECC-9 (New Orleans), Lecture Notes in Computer Science 539, pages 130--139, Berlin, 1991. Springer-Verlag.

....the canonical basis of the Z module of r s (resp. r t ; resp. s t) integer matrices. For 3 dimensional transportation problems see e.g. Vl] and [St, Chapter 14] It is well known that integer programming problems as above can be solved by a method first suggested by Conti and Traverso (cf. [CT]) which resorts to the calculation of suitable Grobner bases (for more information about this method we refer the reader to the survey papers [HT] and [T] More precisely, if I A denotes the toric ideal canonically associated with the matrix A described before, then one needs to find the ....

P. Conti -- C. Traverso, Buchberger algorithm and integer programming, Proceedings AAECC-9 (New Orleans), Springer Verlag LNCS, 539, 130-139, 1991.

....5. A Variant of the Method Using Discrete Relaxations The core of the Integral Basis Method is to perform column substitutions. For this we need to compute irreducible solutions. This can, at least in principle, be accomplished with a combinatorial Buchberger type algorithm, see the papers [CT91,Tho95, UWZ97] and the survey [Wei98] We omit all the details of such an algorithm here because this approach is computationally intractable in general, even for very small instances. Instead, we propose a reformulation technique that is based upon systems that partially describe the underlying problem but for ....

P. Conti and C. Traverso, Buchberger algorithm and integer programming, Proceedings of the AAECC-9, Lecture Notes in Computer Science 539, Springer, Berlin, 1991, pp. 130--139.

....countable set, which, under mild assumptions, is even finite. The key problem of computing function values of Q (which involves solving the second stage problem for various right hand sides) is tackled via a method employing Buchberger s algorithm for computing Grobner bases of polynomial ideals [6]. The integer program is translated into a subalgebra membership problem in a ring of polynomials. The latter is solved by an algorithm for the division with remainder of multivariate polynomials that generalizes the well known division scheme for polynomials in one variable. A Grobner basis of a ....

....once. Moreover, for the Grobner basis computation only algebraic information contained in the second stage is needed, such that the Grobner basis does not depend on the distribution of the random vector . Applying the above method to integer linear programs has been proposed for the first time in [6], see also [26] It yields additional information at a possibly high computational cost that can make it an inefficient method to solve the problem for a fixed right hand side. However, the additional information turns out most beneficial when solving an integer linear program with varying ....

[Article contains additional citation context not shown here]

P. Conti and C. Traverso. Buchberger algorithm and integer programming. In Proceedings AAECC-9 (New Orleans), Lecture Notes in Computer Science 539, pages 130--139, Berlin, 1991. Springer-Verlag.

....linear programs differing only in their right hand sides. For linear programs, solution techniques exploiting this similarity are well known. For integer linear programs this is different, and at the end of the paper we sketch a first method in this respect which is due to Conti Traverso [2] and uses machinery from computational algebra. Its details are also described in [9] Proposition 4.2 provides information on the location of the solution set Psi in terms of lower level sets of the continuous relaxation. Based on the above statements, 9] contains an algorithm for (1.1) ....

....are elaborated in [9] The algorithm s key issue, however, is the efficient computation of Phi(dz i e) Of course, one wants to avoid starting the optimization from the beginning any time a new argument (i.e. righthand side in (1.3) dz i e arrives. Here a solution method proposed in [2] using computational algebra is employed. The integer linear program minfq T y : Wy t; y 2 ZZ m g is translated into a subalgebra membership problem in a suitable ring of polynomials. Lattice points in ZZ m correspond to certain monomials in the polynomial ring. The ring is equipped ....

Conti, P.; Traverso, C.: Buchberger algorithm and integer programming, in: Proceedings AAECC-9 (New Orleans), Lecture Notes in Computer Science 539, Springer-Verlag Berlin, 1991, 130-139.

....P I b are polytopes for all b 2 pos Z (A) For a fixed cost vector c and right hand side vector b 2 pos Z (A) let IP A;c (b) denote the integer program minimize cx : Ax = b, x 2 N n and LP A;c (b) denote its linear relaxation minimize cx : Ax = b, x 2 R n . A number of recent papers (see [CT91], Th94] ST94] have established the existence and construction of the reduced Grobner basis G c ae kernel(A) Z n = ker Z (A) which is a unique minimal test set for the family of integer programs IP A;c ( Delta) The union of all reduced Grobner bases associated with a fixed matrix A is a ....

P. Conti and C. Traverso, Buchberger algorithm and integer programming, Proceedings AAECC-9 (New Orleans), Springer Verlag LNCS 539, 1991, 130-139.

....1 0 for i = 1, k. Since #v i # 1 #y# 1 for i = 1, k, all the summands v i can be written as non negative integral combinations of the elements in H (S ) and hence, y too. # Integral bases can be computed with a combinatorial Buchberger type algorithm, see the papers [CT91, Tho95, UWZ97] and the survey [Wei98] We omit all the details of such an algorithm here because this approach is computationally intractable in general, even for very small instances. Instead, we propose to study systems that partially describe the underlying problem but for which integral bases can be easily ....

P. Conti and C. Traverso, Buchberger algorithm and integer programming, Proceedings of the AAECC-9, Lecture Notes in Computer Science 539, Springer, Berlin, 1991, pp. 130--139.

.... study of the integral vectors in cones [G75] the neighbors of the origin are strongly connected to the study of lattice point free convex bodies [S86] the so called reduced Grobner basis of an integer program is obtained from generators of polynomial ideals that is a classical field of algebra, [CT91]. We refrain within this paper from introducing all these three kinds of test sets, but concentrate on the Graver test set, only. In order to introduce the Graver test set for the family of knapsack problems with varying c 2 Z n and b 2 N, the notion of a rational polyhedral cone and its ....

P. Conti, C. Traverso, Buchberger algorithm and integer programming, Proceedings AAECC-9 (New Orleans), Springer LNCS 539, 130 - 139 (1991).

....i i x j F G Finding an answer to this problem is not only of theoretical interest, but also has an algorithmic impact, because Hilbert bases can be computed by the Buchberger algorithm. The latter algorithm plays an important role for computational algebra in the setting of Grobner bases (see [2], 9] Besides this, there are several generalizations of the problem X i2N 1 x i X i2N 2 x i F that seem interesting. How does the inequality description look like if we replace x i 2 f0; 1g by x i 2 f0; 1; u i g where u i is some natural number (i 2 N) Is there a way to ....

P. Conti, C. Traverso, "Buchberger Algorithm and Integer Programming ", Proc. AAECC 9, Springer Verlag LNCS 539, 130 - 139 (1991).

....entirely in the kernel of the given matrix. We conclude the paper with a section on optimizing with the help of Graver test sets. Here some care has to be taken in the LP case, in particular. Before starting our analysis let us add a few bibliographical remarks. Only in 1991, Conti and Traverso [4] succeeded in actually computing a test set in IP for fixed c by using Buchberger s Grobner basis algorithm [3] The algorithm of Conti and Traverso, however, had a large computational overhead for this application in integer programming and lots of work has been done to improve its performance. ....

P. Conti and C. Traverso. Buchberger Algorithm and Integer Programming. In Proceedings AAECC9, (New Orleans), LNCS 539, Springer-Verlag, 1991, pp. 130--139.

....a test set for the familiy of integer programs IP (b) min c T x : Ax = b, x # Z n associated with a fixed matrix A # Z mn and varying b # Z m . The connection between test sets for integer programming and Grobner bases of toric ideals was first established by Conti Traverso [15]. This algebraic view of test sets is important from an algorithmic point of view. Reduced Grobner bases of toric ideals can be computed by the Buchberger algorithm [13] Reinterpreting the steps of this algorithm as operations on lattice vectors yields a combinatorial algorithm for computing ....

.... . Moreover, the line with endpoints z and z is an edge of the polyhedron conv x # Z n : Ax = Az . ii) Let z 1 and z 2 be two adjacent vertices of conv x # Z n : Ax = Az 1 , then (z 1 z 2 ) gcd(z 1 z 2 ) # G. We mentioned that Conti Traverso [15] established the connection between test sets of integer programs and Grobner bases of toric ideals. The latter objects can be computed by the Buchberger algorithm [13] We discuss below a combinatorial variant of this procedure that allows us to determine a superset of the Graver test set and ....

P. Conti and C. Traverso (1991), Buchberger algorithm and integer programming, Proceedings AAECC9 (New Orleans), Springer LNCS 539, 130 - 139.

....of the right hand side (b; u) but only on the components of u, the upper bounds on z. Our approach uses a simpler algebraic machinery, whereas the one by Thomas and Weismantel is more general. Before starting our analysis let us add some bibliographical remarks. Only in 1991, Conti and Traverso [4] succeeded in computing a test set for (IP ) c;b for xed c by using Buchberger s Gr obner basis algorithm [2] The algorithm by Conti and Traverso, however, had a large computational overhead and since then a lot of work has been done to improve its performance. By fundamental algebraic relations ....

P. Conti and C. Traverso. Buchberger Algorithm and Integer Programming. In Proceedings AAECC9, (New Orleans), LNCS 539, Springer-Verlag, 1991, pp. 130-139.

....has previously appeared in [SW99] Supported by a Gerhard Hess Preis of the Deutsche Forschungsgemeinschaft (DFG) and by a European DONET program. 1 2 1 Introduction Since a few years, there is a strong renewed interest in so called test sets for integer programming problems, see, e.g. [CT91, Tho95, SWZ95b, HS95, TW96, ST97, UWZ97, CUWW97]. This research has not only led to stimulating new insights, but it has also raised quite a few questions. Prompted by the more abstract point of view, these questions often arise as natural generalizations of problems in the design of a primal algorithm to a combinatorial optimization problem. ....

P. Conti and C. Traverso. Buchberger algorithm and integer programming. In H. F. Mattson, T. Mora, and T. R. N. Rao, editors, Applied Algebra, Algebraic Algorithms and Error-- Correcting Codes, number 539 in Lecture Notes in Computer Science, pages 130 -- 139. Springer, Berlin, 1991. Proceedings of the 9th International AAECC Symposium.

....by the Theorem 3 The familly fX v Gamma X v Gamma j v 2 Bs(H; w )g is the reduced Grobner basis of I for the order w on monomials. which is an easy consequence of the previous theorem. 3 Algorithms for toric Grobner bases 3. 1 A good ideal to begin with The usual techniques, used in [4] or [10] is to compute a Grobner basis of the ideals J = X 1 Gamma T a 1 ; X n Gamma T an ; T 1 T 1 Gamma1 Gamma 1; TmTm Gamma1 Gamma 1) or J = X 1 Gamma T a 1 ; X n Gamma T an ; UT 1 : TmX 1 : X n Gamma 1) with Buchberger algorithm and an order ....

P.Conti,C.Traverso "Buchberger algorithm and integer programming", Proc. AAECC-9, LNCS 539,1991,pp. 130-139.

....our objectives is to supply users of Grobner bases software with some new algorithmic tools. The Grobner minded reader will notice a surprising interplay between ffl Grobner bases for commutative polynomials (the classical version; see e.g. 4] ffl Grobner bases for integer programming (as in [6], 21] 25] ffl Grobner bases in the ring of differential operators (as in [15] 23] 2. A generating function for feasible points We fix a linear map T : N n N d as in (1.1) For each ff 2 N d the fiber T Gamma1 (ff) f u 2 N n : Au = ffg is a finite set. The integer ....

.... satisfy the commutation relations x i x j = x j x i ; i j = j i ; i x j = x j i if i 6= j; and i x i = x i i 1: In the commutative polynomial subring Q[ 1 ; n ] of An we consider the toric ideal I A : Gamma u Gamma v : Au = Av Delta : Recall from [6], 21] and [25] that the integer programming problem can be solved by normal form reduction modulo the Grobner basis of I A with respect to . 2 For any ff 2 Q d we introduce the linear differential operators Z i (ff i ) n X j=1 a ij x j j Gamma ff i for i = 1; d: The ....

P.Conti and C.Traverso, Buchberger algorithm and Integer programming, Proceedings of AAECC-9, Springer Lecture Notes in Computer Science (1991), 130--139.

....c yields a test set for the familiy of integer programs IP (b) minfc T x : Ax = b; x 2 Z n g associated with a xed matrix A 2 Z m n and varying b 2 Z m . The connection between test sets for integer programming and Gr obner bases of toric ideals was rst established by Conti Traverso [15]. This algebraic view of test sets is important from an algorithmic point of view. Reduced Gr obner bases of toric ideals can be computed by the Buchberger algorithm [13] Reinterpreting the steps of this algorithm as operations on lattice vectors yields a combinatorial algorithm for computing ....

....n : Ax = Az g. Moreover, the line with endpoints z and z is an edge of the polyhedron convfx 2 Z n : Ax = Az g. ii) Let z 1 and z 2 be two adjacent vertices of convfx 2 Z n : Ax = Az 1 g, then (z 1 z 2 ) gcd(z 1 z 2 ) 2 G. We mentioned that Conti Traverso [15] established the connection between test sets of integer programs and Gr obner bases of toric ideals. The latter objects can be computed by the Buchberger algorithm [13] We discuss below a combinatorial variant of this procedure that allows us to determine a superset of the Graver test set and ....

P. Conti and C. Traverso (1991), Buchberger algorithm and integer programming, Proceedings AAECC9 (New Orleans), Springer LNCS 539, 130 - 139.

....x # # k[x] is a standard monomial of in c (I A ) if and only if # is the unique optimal solution to the integer program IP A,c (A#) By Corollary 2, there is a bijection between the standard monomials of in c (I A ) and the elements of the monoid pos Z (A) The Conti Traverso algorithm. In [9], Conti and Traverso gave an algorithm to solve integer programs using Grobner bases of toric ideals. A Grobner basis with respect to c, of the toric ideal I A , is any finite subset H of I A such that in c (I A ) #in c (f) f # H#. A Grobner basis H is reduced if it has the additional ....

Conti, P., and Traverso, C.: `Buchberger Algorithm and Integer Programming', in H. F. Mattson, T. Mora, and T. R. N. Rao (eds.): Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science 539, Springer-Verlag, 1991, pp. 130--139.

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P. Conti and C. Traverso. Buchberger algorithm and integer programming. In AAECC-9, volume 539 of LNCS, pages 130--139. Springer, 1991.

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Pasqualina Conti and Carlo Traverso. Buchberger algorithm and integer programming. In Applied algebra, algebraic algorithms and errorcorrecting codes (New Orleans, LA, 1991.

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P. Conti, C. Traverso, Buchberger algorithm and integer programming, Proceedings AAECC-9 (New Orleans), Springer LNCS 539, 130 - 139 (1991).

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P. Conti and C. Traverso, Buchberger Algorithm and Integer Programming, Proceedings AAECC-9 (New Orleans), Springer Verlag LNCS 539 (1991) 130-139.

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