REVUE FRANÇAISE D INFORMATIQUE ET DE

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1 REVUE FRANÇAISE D INFORMATIQUE ET DE RECHERCHE OPÉRATIONNELLE, SÉRIE ROUGE G. MITRA D. B. C. RICHARDS K. WOLFENDEN An improved algorithm for the solution of integer programs by the solution of associated diophantine equations Revue française d informatique et de recherche opérationnelle, série rouge, tome, n o 1 (19), p. -. < 1 > AFCET, 19, tous droits réservés. L accès aux archives de la revue «Revue française d informatique et de recherche opérationnelle, série rouge» implique l accord avec les conditions générales d utilisation ( Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

2 R.I.R.O. ( e année, N R-l, 19, p. -) AN IMPROVED ALGORITHM FOR THE SOLUTION OF INTEGER PROGRAMS BY THE SOLUTION OF ASSOCIATED DIOPHANTINE EQUATIONS by G. MITRA, D. B. C. RICHARDS and K. WOLFENDEN (*) Résumé. Valgorithme du type «Cutting plane» qui est élaboré ici semble avoir quelques avantages sur les algorithmes actuels du même type. Cet algorithme se sert d'un autre, le «Positive Diophantine», qui donne la solution d*une équation diophantine à variables non négatives. De ceux-là se développe une nouvelle méthode, méthode directe, qui donne les solutions des programmes avec des valeurs entières. Trois exemples détaillés illustrent la technique, ^algorithme «Positive Diophantine» se trouve à Vappendice. INTRODUCTION In the course of gênerai studies on techniques of integer programming an algorithm of the cutting plane type has been developed which appears to offer certain advantages over existing algorithms of the same type ([1, []). This algorithm makes use of another, «Positive Diophantine», for the solution of diophantine équations in non-negative variables. Together these have led to the development of a further method, a direct method similar to the technique outlined in []. 1. THEORY 1.1. The basic problem Consider the problem of maximizing *o = «oo + Ê a oj{ Xj) (1.1.1) (1) University of London Institute of Computer Science.

3 G. MITRA, B. RICHARDS, K. WOLFENDEN subject to and n X UijXj < a i i = 1,,..., m (1.1.) x, ^, xj = mod (1) j =,1,,...», where for convenience it is assumed that ail a^ = mod (1). Introducing slack variables x t = mod (1), x t >, i = n + 1,» +,..., n + m, and adding a set of trivial équations Xj = ( Xj),j = 1,,..., n, leads ta the Tucker-Beale System = + E.. n (1.1.) *i = ^ + S ö '( ^j) I = «+ 1,» +,..., H + W, where S SJ. is the Kronecker delta. 1.. The continuous solution Let C(Â) represent the continuous optimum solution to this problem as obtained by a simplex algorithm. This solution is contained within the form Dx = Dx t =,. o (1..1) where B dénotes the set of indices of the vectors forming the current basis, namely, the basis of the optimal program C(Â). The coefficients D> the modulus of the determinant of the current basis, and â tj, the éléments of the matrix Â, are all intégral. Setting the non-basic variables x p to zero, the continuous optimum solution is x ( = O /D. Let I(Â) dénote the optimal integer program of the stated problem. If a i & mod (D) for any /, 1,,..., m + n, then 1() & C(A).

4 SOLUTION OF INTEGER PROGRAMS 9 1** Catting plane steps Let e be the index of any row of the system (1..1) such that a ep & mod (D) for at least one p $B. For such a row define the fractional éléments and D D pib, < /o < 1. sothat As in [] this leads to the introduction of the reduced inequality 1, (1..1) fo< YsfpXp -.) which must be satisfied by any feasible solution to the problem. However, if the fractional parts are expressed as ratios of integers with common denominator Z>, then under certain conditions a stronger inequality can be constructed. From (1..1) we can obtain non-negative integers d Oi d p such that f _ do " and substituting in (1..). = -, o d 9 d p < D, (1..) (1..) Of course, this is just one of a series of parallel «cuts» given by do + rd^ X d v x v r =, 1, %..., (1..) P&B and such that the larger the value of r the deeper the eut into the convex région (1.1.). It would appear désirable, therefore., to be able to détermine the largest possible value of r such that the corresponding eut does not exclude any feasible lattice point. However, the best that can be done is to find the minimum value of r for which the diophantine équation do + rd = X P$ has a non-negative integer solution in the variables x p. Note that in the case of a primary eut (r = ) the diophantine équation always admits of solution in integer variables unrestricted in sign. Formally, the problem now is to find minimum r and x p = mod (1), such h thath (1..)

5 G. MITRA, B. RICHARDS, K. WOLFENDEN A dynamic programming solution has been proposed in [] which, though conceptually elegant, is hardly computationally efficient. An alternative approach is provided by the algorithm «Positive Diophantine» (Appendix). The aim is to locate lattice points on the finite parallel planes (1..) within the bounds ^ x D < - = In most cases where cuts with r > (secon- L d dary cuts) have been found Jto exist, the convergence to I(A) has been more rapid than with primary cuts. Computationally worthwhile improvement has been observed in a number of test problems (Table 1), only two taking more itérations than with Gomory's method. 1.. Direct détermination of I(Â) from C(Â) It is possible to proceed from C(Â) to I(Â) in the manner of [] and at each eut génération stage to apply the process outlined in the previous section. However, the additional pivoting involved can be avoided by taking advantage of the search technique of Section 1.. Let X = { x t x ( = â i /D, i çb ;x t =,i $ } dénote the solution C(Â) of the continuous problem. Then corresponding to an optimum integer solution X 1, there exists a non-negative integer vector of n components x p = G mod (î), p i B, such that X 1 can be expressed ([]) as X 1^ {x;\xl = â io fd+ Z & ip ID)( x p ),i <=B; x! = x p,i = p $B X (1..1) where of necessity x\ = mod (1) and x\ ^, i = 1,,..., n + m. From(lAl) it follows that to obtain the integer program directly we need to détermine the components of this ail important «-vector of the variables x p, p $B. To do this we can use the objective row of  without extracting the eut and investigate the parallel hyperplanes corresponding to different values of r until a feasible lattice point can be located on one of them. If, from the objective row of the matrix  we extract the congruence and rewrite it in the form (D) d o + rd= X â Op x p x p = mod (1), x, > (1..)

6 SOLUTION OF INTEGER PROGRAMS 1 then, it is to be noted? (1..) poses the same problem as (1..), namely, solution in positive integers of a diophantine équation with positive coefficients (â Op are all non-negative integers, since the optimal tableau is of necessity dual feasible (*)). For a fixed value of r, (L.) defines a finite plane in the w-space of x p, p $B, and all the lattice points on this plane are generated by the search «Positive Diophantine». These in turn are substituted in the relation (1..1) and the results x\, i B, are checked for non-negativity and integrality. Assuming C(Â) is not dual degenerate, that is, the objective row of A does not contain any zero element, an algorithm for obtaining I{Â) from C(Â) is outlined below. 1. Extract (1..) from (1..1) and set r =.. Apply the algorithm «Positive Diophantine» to explore whether there exist x p = mod (1) and x p >, p $ B, which satisfy the equality of (1..).. Is the present hyperplane exhausted? If yes then r : = r + 1 ; go to step.. x p >,p B, are the components of an integer vector satisfying (L.). Does this vector substituted in (1 Al) satisfy the integrality and non-negativity requirements of X 1! If yes then go to.. Go to.. Output the optimal integer solution and stop. Note that the algorithm can be simply modified to produce alternative optima if they exist. Feasibility or otherwise of the current program for each successive value of r can likewise be established with little extra effort.. EXAMPLES.1. Generaüzed eut Consider the problem (taken from []) : Minimize x subject to *! + lx > 1 x A + 1x > (.1.1) Xj == mod (1) and Xj > = 1,. (1) For the time being we assume that öo P > for all p&b. The problem of dual degeneracy is dealt with in Section..

7 G. MITRA, B. RICHARDS, K. WOLFENDEN Writing the tableau in the Tucker-Beale form and applying the dual simplex pivot rules we obtain the continuous optimum as follows : Itération x X * 1 1 Note : x dénote slacks ; pivot element is starred. Itération 1 D = = x x x x t x * 1 1 9* Itération D = = 9 x x A x x 1 * XA Continuous optimum. Using the objective row to generate the eut, we have by relation (1..) + 9r ^ 1x + * (.1.) Taking r = for this and all the subséquent cutting planes (Gomory's method []), the optimum integer solution was obtained after pivots steps.

8 SOLUTION OF INTEGER PROGRAMS However, applying «Positive Diophantine» to (.1.) as described in Section 1. we obtain min(r) =, and append the corresponding generalized eut to Itération. Itération D =9 x. x X, x x x *1 ( + x 9) * Note : A random choice is made to break the tie for the sélection of pivot column. Itération D = x Xi x x x (+ X ) * Note : Second row is used for eut extraction. Applying «Positive Diophantine» we obtain min (r) = for which there exists a solution in positive integers for x and s t. Itération x x

9 G. MITRA, B. RICHARDS, K. WOLFENDEN Note : The optimum integer solution obtained in itérations is _ mm x = ^~ = 19 1= 1. n -^~ =.. Direct method Consider the continuous optimum tableau of the previous problem (.1.1). D =9 X X x *1 x * We extract the équation + 9r = 1* + x (..1) from this tableau as explained in Section 1. (note that this is the same as the eut extracted from the objective row in the first cutting plane step of the last section). Applying the algorithm of Section 1. to équation (..1) we obtain min (r) = and the corresponding solution + (9) = 1() + (), i.e. x =, JC =. Substituting this in the continuous tableau as in (1..1) we have 9(AT ) = () () = (9) = mod (9) 9(x x ) () () - (9) = mod (9) ^ 9(x ) = + 1() + () - (9) = mod (9) ^ 9(x ) = + 9() + - (9) = mod (9) > 9(x ) = + + 9() - (9) = mod (9) ^ and the solution I(A) has been obtained directly from C(Â).

10 SOLUTION OF INTEGER PROGRAMS Consider next the application of this direct method to another problem from [] : Minimize x subject to x t + x x > 9 = mod (1), x t x + x > 9 = 1,,. Itération X X * 1 Itération 1 D = ~ "V* ^^_ "V* JC JÇ x 11 x 1 Itération D = 1 X 1 * Xg x x x x x 1 1 9* 19 1

11 G. MITA r B. RICHARDS, K. WOLFENDEN Itération D =9 X. X$ X x Continuous optimum Extracting the équation (1..) from the objective row, 9 + 9r = 1* + 9x + x, and applying the algorithm of Section 1., we obtain the minimum r for which there exists a solution, namely min (r) =. Then i.e. 9 + (9) = 1() + 9() + (1), x = x =, Substituting in the above tableau : 9(x ) = 1 () 9() (1) - 9* = () + 19() + (1) = 9(x ) () + 1() + (1) - 9(x ) = () + 9() + (1) - 9(x ) - + 9() (x )= + +9()+ = 9(x )= (1) = (9) s mocl (9) (9) = mod (9) (9) s mod (9) 1(9) = mod (9) (9) = mod (9) (9) = mod (9) 1(9) s mod (9) which is the solution I(A) to the integer problem. Lastly, to bring out the relationship between the proposed method and dynamic programming, we consider a cargo-loading problem taken from [l]. Maximize xh subject to - x -f x x t -f - 1x + 1x x t x, x + x + x + x + 9x 1x + 1x + 1x + x + x ^ 1, and x u x,..., x = mod (1).

12 SOLUTION OF INTEGER PROGRAMS Using the reduced tableau, that is, omitting the unit rows, we have Itération D = 1 X1 x x x x * Itération 1 Z>= x t x -x s _* -*s -x* x We now investigate by ' * Positive Diophantine ' * the problem of minimizing r subject to non-negative intégral solution of + r() = I9x x + x + JC + x + ;t + x + x + 9x 9. It is found that no such solution exists for r =, 1, whereas for r = the solution (,,,,,, 1, ) is unique. Substituting in the constraint équation we have x = 1 (1) = ^ mod (). Next, r = has the solution (,,,,,,, 1) but again the integrality xequirement of the constraint équation is not satisfied, x s = 1 (1) = 99 & mod (). The cyclic process of incrementing r, determining lattice points and testing the constraint is continued until r. The solutions of the diophantine équation are now (,,,,,,,), (1,,,,,,,), (,,,,,,, ). The first two solutions do not satisfy the integrality condition for x but the last one does, x = 1 () =9- mod (). Hence x =, x 1 = x =... = x 1 = is the optimum solution to the problem with x =.

13 G. MITRA, B. RICHARDS, K. WOLFENDEN.. Results and Conclusions In ail some problems were tackled by both the optimum eut method and Gomory's Method of Integer Forms on the ICT Atlas at the Institute of Computer Science. These problems, taken from a number of sources, have all been rated as difficult integer programming problems in some sense or other. Several variants of the two main Fortran programs were written to accommodate different eut generator sélection rules. Complete results for a matched pair of programs are given below; overall exécution times for the optimum eut method and Gomory's Method were. secs and. secs TABLE l j PROBLEM' Np. x-zuwlcrn NO. OF ITERATIONS TO INTEGER SOLUTION (including the continuous stage) Constraints Variables Gomory's Method Optimum Cut Method ' >

14 ALGORITHMES DE CALCUL LINEAIRE 9 respectively, the latter including a problem which remained unsolved after the imposed limit of itérations. Although our expérience with both the optimum eut method and the related direct method is limited (program development with the direct method is as yet incomplete) the following points should be noted. 1) The D-number can become too large for single length arithmetic even before the continuous optimum C(Â) is reached, thus inhibiting the application of either method. To combat this an adaptative eut génération technique [] which attempts to restrain the growth of D might be applied. ) In the direct method, if the continuous optimum solution is dual degenerate then there will exist an infinity of solutions for the diophantine équation extracted from the objective row, namely d + rd = a 1 x x + â 1 x â On x n (..1) For if â Oi for at least one i, 1 < i ^ n, x % can assume any non-negative intégral value. This difficulty might be resolved by holding the already determined components of a solution to (..1) at their current values and varying the parametric values of the indeterminate components (corresponding to â Oi ) in a lexicographically ordered search on the successive équations. APPENDIX. THE ALGORITHM "POSITIVE DIOPHANTINE" Consider the congruence = a mod (D) (A.l) where a f = mod (1) and a f >, x t E= mod (1) and x t >, i = 1,,..., n. For convenience let us assume that the coefficients are ordered, so that Let the diophantine équation a t ^ oc < a... < a n. «i*,. = a + rd = ) n (A.) define the problem P n. Then set x n - L so that the remaining JCj must L a - J satisfy

15 G. MITRA, B. RICHARDS, K. WOLFENDEN defining the problem P n -t. Répétition of the procedure leads to <& a defining the problem P u and we define If at any stage > fc is zero, the congruence (A.1) is satisfied by the x t already set for i > k + 1, and x t for i < fe. If the entire séquence i\-, / = «, «,..., 1, is produced and <I> ^, then a new subsequence must be defined, starting at the last i for which x t >. Suppose this was i = k, then reset x k ~ x k ^n( i ^fc-i ~ ^fc-i + a k afl d proceed with the new subproblems P i9 i = k? k,..., 1. Whenever O f =, the corresponding «-vector is a lattice point on the hyperplane defined by (A.). The recursive process can be continued until the search over the finite hyperplane is complete, that is, the components x ni x n^u..., x are reduced to zero. REFERENCES [1] BELLMAN, R. E. and DREYFUS, S., Applied Dynamic Programming (Princeton University Press, 19), pp. -1. [] FINKELSHTEYN, Yu. Yu. ; Additional Restrictions for Problems of Integer Linear Programming, Engineering Cybernetics (Translated from Russian), May-June 19, pp. -9. [] GOMORY, R. E., An Algorithm for Integer Solutions to Linear Programs, Recent Advances in Mathematical Programming, ed. Graves, R. L. and Wolfe, P. (McGraw HUI, 19), pp. 9-. [] GOMORY, R. E., An All-Integer Integer Programming Algorithm, Industrial Scheduling, ed. Muth, J. F. and Thompson, G. L. (Prentice Hall, 19), pp [] GOMORY, R. E., On the Relation between Integer and Non-Integer Solutions to Linear Programs, Proceedings of National Academy of Sciences (U.S.), vol. (19), pp. -. [] THOMPSON, G. L., The Stopped Simplex Method Basic Theory for Mixed Integer Programming, Integer Programming, Revue Française de Recherche Opérationnelle, vol (19), pp