Aggregation of a class of systems of integer-valued equations

U.S.S.R. Comput. ~~ths~~th. Phys. Vol. 18,~~. 86-91 0 Pergamon Press Ltd. 1979. Printed in Great Britain.

A~REGATIO~ OF A CLASS OF SYSTEMS OF ~TEGER-VALUED EQUATIONS* DZh. A. BABAEV and K. SH. MAMEDOV Baku (Received 12 July 1976; revised23 June 1977) THE NON-NEGATIVE integer solution of a linear equation of a special type is shown to be unique. On this basis we devise a method of constructing an equation, equivalent to a system of equations of a certain class with integer-v~ued left-hand sides. 1, It is well known that the difficulties of solving problems of integer-valued programming increase rapidly with the dimensionality of the problem, i.e., with the number of unknowns and constraints. Hence it is extremely important to find ways of reducing the dimensionality of such problems. Of great interest from this point of view is the possibility of constructing one equation, equivalent to a given system of equations with non-negative integral solutions. We shall refer to the process of construct~g this single equivalent equation as aggregation of the system of equations. Various methods have so far been proposed for aggregating systems of linear equations with integer coefficients [l--3] and systems of nonlinear integer-valued equations [4]. Yet all the methods so far proposed lead to an equivalent equation with coefficients having extremely / large values. When seeking the integral solutions of an equation, in order to avoid the possibility of losing solutions we have to operate with exact values of the coefficients. If the coefficients are very large, their exact representation in a computer involves serious technical difficulties. Hence only equivalent equations with relatively small integral coefficients, capable of exact repre~ntation on the computer, are of interest for use in practical computations. In the present paper we show that the equation cp==2 (2n--2n--k) (&-1) kx;i

=o

(1)

has a unique non-negative integral solution, and we use this result for the aggregation of the system of equations f,(X)=& where X=(CC~, . . , , 5,) integral values.

i-1,2,

and f<(z), i-1,2,.

5%. vjkhisl. Mat. mat.. Fiz., 18,3,614-619,1978.

*. . , m, . ., m,

X&i?, for X E C C Rn , take non-negative

(2)

8’7

Systems of integer-valuedequations

Our present method of aggregation is compared in Section 4 with previously published methods. 2. We have:

Equation (1) has a unique non-negative integral solution xk = 1 for any k, k = 1,2, . . . , n. Proof Obviously, any other solution must contain xP = 0 and xq > 1. Let &={+~:,>1).

K,={kl%=l),

K,=(kjs=O),

Then, Eq. (1) takes the form Q= c

(P-29

(X&--l) - x

AWL2

(27z-2’+“) =(I.

kid&,

Let

Here, I 82 1denotes the number of elements in the set fz. Then,

@pz(A&--N*) 2”+

z

c Y&P-“,

Pk-

Rex Kg

kK,

or @=@*t#O-@~, where @,= (N,-N0)2”,

CD0=

IS

kEK(

2”-k,

@, = 2

&‘2”-‘.

k.z&

Clearly, at most No and N2 terms of the type 2s are to be found in the functions @pgand @2 respectively. Let us show that, given any integers ~~3’1, k&, we have @=@,l+@ocD,zo. Consider two cases. &sea. Assume that Yk=i for &din. Then, all the terms appea~ng in +o and @2 are distinct, since KofX2=%, and among them only one is a least term 2s < 2n. Consequently, where A is an integer. Hence @ 4 0. 1. In this case a2 consists of N2 terms of the type 23, among which some terms are identical. But they are all different from the terms appearing in @I and Qo.

DZh. A. Babaevand K. Sh. Mamedov

88

In view of the relation 2”+25=25+1

(3) . . . , we obtain powers of 2 appearing in

on adding the identical terms in a2 in groups of 2,4,8, Qp1and @o.

To obtain Q = 0, the terms appearing in al and @o must cancel when like terms are grouped. This can only happen by virtue of the terms of Q. According to (3) to annihilate every term of a.0 we require at least two terms of @2, i.e., we have to satisfy the condition fV2 > 2No. Hence it follows that N,--N&O. Consequently, cP1> 0, and for the annihilation of every term of @I, we also require at least two terms of @2. Then, to obtain Cp= 0, we have to satisfy the condition Nt>2No+2 (N2-No) =2Nz. This condition can be satisfied only with N2 = 0, i.e., when the set K2 is empty. Hence, if i.e., there exists k E K2, then @ $ 0. The theorem is proved. A more general theorem can be proved: Theorem 2

The equation II

c

a>i,

(un-an-k)(x$,-i) -0,

an integer,

k-1

has a unique non-negative integral solution xk = 1 for any k, k = 1,2, oo o , n. The proof is similar to the proof of Theorem 1. 3. Corollary. Let X= (xi, . , . , x,) and for X E G C Rn, let the functions f,(z), i=l, take non-negative integral values. Then, system (2) is equivalent to the equation 2,-**, n,.

cm

(p+p-k

=G,

1(h(X) -1) =O,

(4)

k-1

where a > 1 is an integer. The proof follows immediately from Theorem 1o 4. Example 1. In many problems of integer-valued programming (e.g., in allocation, assignment, the commercial traveller, and reconstruction problems etc., see [5] , we encounter constraints of the type si=x,i+. . .+5in=1, .

.

.

.

.

.

.

.

(5) Ism=2,,i+.

. .+x,.=1,

xji 2 0 are integers, i=l, 2, . . . , m, j=i, 2, . . . , n. Methods are given in [ 1,2] for aggregating systems consisting of two linear integer-valued equations with non-negative coefficients and

right-hand sides. For a system containing more than two equations, on repeatedly applying the method of [ 1] or [Z] , we can construct an equivalent equation. As was pointed out in [2] , the method there employed leads to an equivalent equation with smaller coefficients. Using this fact, we shall employ the method described in [2] to aggregate the system of [5], taking the case m = 10. After nine steps of pairwise aggregation of the equations, we obtain

The corollary of Section 3 gives a method for aggregating system (5). Whereas, in the methods of [ 1,2] , the equivalent equation is constructed by successive steps, here it can be written directly, since its coefficients are known in advance (see (4)). In order to write Eq. (4) for the system (5), it is sufficient to note that

%f=&*

k-l,

2,. . . , m.

On substituting this in (4) putting a = 2, and taking the unattached terms over to the right-hand side, we obtain

Evidently, the maximum coefficient in (7) i.e., when the method of the present paper is used for aggregation, is almost one tenth, and the right-hand side less than one third, the corresponding values in (6) i.e., when the method described in [2] is used for aggregation. It should be mentioned that, as the number m of equations increases, this difference increases rapidly. Example 2. Let us reduce a system of linear inequalities with different coefficients to a system with non-negative coefficients. Given the system

where aii and bi are arbitrary numbers, and xi = 1 or 0. In the case of non-negative coefficients ajj, the study of systems of the type (8) is greatly simplified; for instance, it is easy to find one or more solutions of the system. It is therefore of interest to consider the possib~ity of reducing (8) to an equivalent system with non-ne~tive coefficients (see [6] ). Let J1, J2, J3 be disjoint subsets of indicesi such that J~Ui&J1.,= {1,2, . . . , n}, and moreover, let aij>O for j=JI for all in {I, 2, . . . , m}, ai+0 for j=Jz for all for i EJ3 there exist il and i2 such that aisj>O and Ut&O. Then, &={I, 2,. * . , m}, for i E J3, we substitute

U
i-l,

2, . . . , m,

(9)

DZh. A. Babaev and K. Sh. Mamedov

90

where a([=max (0, Qj) , afj” =min system of equations

(0, a<)), Yj: zj=l

Yi-Zj=O*

at

0. Further, we add to (8) the

jE1s*

Let us change the notation for the xi, j EJ2, to zj. If, after this, we introduce the change of variables zi = 1 - z+, then (8) transforms to the system

(11) By hypo~esis, U+j>O for j=f,, ail>0 for J’EJ~, aij
Here,

I,=‘,

&‘=(l,

On introducing the substitution (9) and putting x4 = 24, we get -2z,+y,--z,+y,+2y,-z&-l yl-2zsz,+3Y,--2z,+2Y,~l, yz+2y,--2z,+ y,--3z,G - 1, yz-zz=o, Yl-si=o, Y,-z,=O, ye--ze=O, 7Jj,

ys-zs=o,

y7-&=O,

Zj=l or 0,

j=l,

2 7 q-s, 7.

The substitu~on Zj = 1 - EI~gives

(12)

91

Systems of integer-valued equations

Equations (12) satisfy the conditions of the corollary. This system of six equations can be replaced by the equivalent equation (4) which, in the present case, takes the form (a = 2)

32 (yi+u,) +48(y,+uz) +62(y,+u,)+63(y7+u,)

+56( ~3-I-us) +60 (ys+us) ~321.

(13)

Application of the method described in [2] for the aggregation of system (12) gives 32 (y,+u,) +48(yz+Q f56 (Y~+G) +60(y,+us) +86 (ye+@ +I27 (Y~+u,) =409. The right-hand side of this equation is larger than in (13). It was remarked above that, if the number of equations in system (12) is increased, this difference increases rapidly. It is clear from (4) that the coefficients of the equivalent equation depend only on the number m of equations in the system and on the quantity a. Naturally, when we use the corollary of Theorem 2 to obtain the least values of the coefficients we have to take a = 2, A method of aggregation of systems of linear integer-valued equations was also described in [3], though it is only applicable under the extra condition that the unknowns are upper-bounded by given numbers. It is easily shown that, when aggregating systems of type (5), even when this condition is satisfied, the method described in [3] is much less satisfactory than the method of the present paper. Translated

by D. E. Brown

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1897.

2.

GLOVER, F., and WOOLSEY, R. E. D., Aggregating diophantine equations, Zh. operat. Res., 16, No. 1, l-10,1972.

3.

PADBERG, M. W., Equivalent knapsack-type formu~tions of bounded integer linear programs: an alternative approach, Naval Res. Logist. Quari, 19,No. 4,699-708,1972,

4.

MAMEDOV, K. Sh., Problem of aggregating a system of non-linear diophantine equations, D&l. Akad. Nauk Azerb SSR, 33, No. 2,3-S, 1977.

5.

KORBUT, A. A., and FINKTEL’SHTEIN, Yu. Yu., Discrete programming (Diskretnoe programmirovanie), Nauka, Moscow, 1969.

6.

GARFINKEL, R. S., and NEMHAUSER, G. L., A survey of integer programming emphasizing computation and relations among models, In: Math. Pragt., Acad. Press, New York, 77-155,

7.

BABAEV, Dzh. A., Method of hyperspheres for solving problems of Boolean pro~~rn~g, Mat. mat. Fiz., 16, No. 6,1414-1426,1976.

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Zh. vychisl.