Operations Research Letters 34 (2006) 564–568

OperationsResearchLetters

www.elsevier.com/locate/orl

Carathéodory bounds for integer conesFriedrich Eisenbrand∗, Gennady Shmonin

Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany

Received 16 December 2004; accepted 22 September 2005Available online 8 November 2005

Abstract

We provide analogues of Carathéodory’s theorem for integer cones and apply our bounds to integer programming and tothe cutting stock problem. In particular, we provide an NP certificate for the latter, whose existence has not been known so far.© 2005 Elsevier B.V. All rights reserved.

Keywords: Carathéodory’s theorem; Integer cone; Integer programming; Cutting stock; Bin packing

1. Introduction

The conic hull of a finite set X ⊂ Rd is the set

cone(X) = {�1x1 + · · · + �t xt | t �0;

x1, . . . , xt ∈ X; �1, . . . , �t �0}.Carathéodory’s theorem (see, e.g. [7]) states that if b ∈cone(X), then b ∈ cone(X), where X ⊆ X is a subsetof X whose cardinality is bounded by the maximumnumber of linearly independent points of X and thusis bounded by d.

The integer conic hull of a finite set X ⊂ Rd is theset

int_cone(X) = {�1x1 + · · · + �t xt | t �0;

x1, . . . , xt ∈ X; �1, . . . , �t ∈ Z�0}.

∗ Corresponding author.E-mail addresses: eisen@mpi-inf.mpg.de (F. Eisenbrand),

shmonin@mpi-inf.mpg.de (G. Shmonin).

0167-6377/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.orl.2005.09.008

The analogous question here is the following: givenb ∈ int_cone(X), how large is the smallest subset X

of X such that b ∈ int_cone(X)?It is easy to see that one cannot give an upper bound

for the cardinality of a smallest subset X in terms ofthe dimension d. For example, if X ⊂ Zd consists ofthe vectors xij = 2iej + ed for i = 0, . . . , n − 1 andj = 1, . . . , d − 1, where ej is the jth unit vector, thenall these vectors are needed to represent the vectorb ∈ Zd with bj = (2n − 1), j = 1, . . . , d − 1, bd =n(d − 1). Indeed, b = ∑n−1

i=0∑d−1

j=1 xij is the uniquerepresentation.

The size of an integer a ∈ Z is the number of bitswhich are needed to represent a in binary encoding. Itis convenient to define it as size(a)= log(|a|+1). Thesize of a vector v ∈ Zn, denoted size(v), is the sum ofthe sizes of its components. In the example above, wehave size(b) = n(d − 1) + log(n(d − 1)), and hencethe number n(d − 1) of nonzero terms that are nec-essary in order to provide b as a result of an integerconic combination can be arbitrary large compared

F. Eisenbrand, G. Shmonin / Operations Research Letters 34 (2006) 564–568 565

to the dimension; this number has the same order ofmagnitude as size(b).

In this note, we show the following integer ana-logues of Carathéodory’s theorem. The first theoremprovides polynomial bounds in the dimension and thelargest size of a component of a vector in X.

Theorem 1. Let X ⊂ Zd be a finite set of integervectors and let b ∈ int_cone(X). Then there exists asubset X ⊆ X such that b ∈ int_cone(X) and thefollowing holds for the cardinality of X:

(i) If all vectors in X are nonnegative, then|X|�size(b).

(ii) If M = maxx∈X‖x‖∞, then |X|�2d log(4dM).

In the second theorem, we suppose that X is closedunder convex combinations. The latter means that ev-ery integer point in the convex hull of X also belongsto X. The theorem provides then an exponential boundin the dimension.

Theorem 2. Let X ⊂ Zd be a finite set of integer vec-tors which is closed under convex combinations andlet b ∈ int_cone(X). Then there exists a subset X⊆X

of cardinality |X|�2d such that b ∈ int_cone(X).Furthermore, if b = ∑

x∈X �xx, �x ∈ Z�0 for allx ∈ X, then there exist �x ∈ Z�0, x ∈ X, such thatb = ∑

x∈X �xx and∑

x∈X �x = ∑x∈X �x .

As an application of these results we prove that thecutting stock problem has an optimal solution whosebinary encoding length is polynomial in the inputlength and that the cutting stock problem with a fixednumber of item sizes has an optimal solution with afixed number of patterns.

1.1. Related work

A finite set of vectors X is called a Hilbert ba-sis, if each integer vector b ∈ cone(X) is a non-negative integer combination of elements in X. Cooket al. [2] provided the following integer analogue ofCarathéodory’s theorem: if cone(X) is pointed and ifX ⊆ Zd is an integral Hilbert basis, then for every b ∈cone(X) there exists a subset X ⊆ X with |X|�2d−1such that b is an integer conic combination of vec-tors in X. Sebo [8] improved this bound to 2d − 2.

Bruns et al. [1] have shown that the bound d is notvalid. If X is an integral Hilbert basis but cone(X) isnot pointed, then an upper bound in terms of the di-mension d cannot be given [2]. Cook et al. [2] alsoconsidered integer programs min{1Ty |Ay = b, y ∈Zm

�0}, where A ∈ {0, 1}d×m and ATx�1 has the in-teger rounding property. The authors showed that, ifthere exists an optimal solution to such an integer pro-gram, then there exists an optimal solution with atmost 2d − 2 nonzero coefficients.

2. Proofs of the theorems

To prove the above stated assertions, we exploit thefact that, if the sum of some vectors in X can be ex-pressed as the sum of other vectors in X, then wecan eliminate one (or more) vector used in the integerconic combination. A similar idea has also been ap-plied by Sebo [8] to prove his conjectures for Hilbertbases in particular cases.

Lemma 3. Let X ⊂ Zd�0 be a finite set of non-

negative integer vectors and let b ∈ int_cone(X).If |X| > ∑d

i=1 log(bi + 1), then there exists a proper

subset X of X such that b ∈ int_cone(X).

Proof. Let b = ∑x∈X �xx with �x > 0 integer for all

x ∈ X. Clearly,∑

x∈X x�b for any subset X ⊆ X.This implies that the number of different vectors whichare representable as the sum of vectors of a subset X ofX is bounded by

∏di=1(bi +1). If 2|X| >

∏di=1 (bi +1),

there exist two subsets A, B ⊆ X, A �= B, with∑x∈A x =∑

x∈B x. Consequently, there exist two dis-joint subsets of X, A′ = A\B and B ′ = B\A, with∑

x∈A′ x = ∑x∈B ′ x. Suppose that A′ �= ∅ and set

� = min{�x : x ∈ A′}. Then we can rewrite

∑x∈X

�xx =∑

x∈X\A′�xx +

∑x∈A′

�xx

=∑

x∈X\A′�xx +

∑x∈A′

(�x − �)x + �∑x∈A′

x

=∑

x∈X\A′�xx +

∑x∈A′

(�x − �)x + �∑x∈B ′

x

=∑x∈X

�xx,

566 F. Eisenbrand, G. Shmonin / Operations Research Letters 34 (2006) 564–568

where �x = �x if x ∈ X\(A′ ∪ B ′), �x = �x + � ifx ∈ B ′ and �x = �x − � if x ∈ A′. Thus, �x �0 for allx ∈ X and at least one of the �x , x ∈ A′, is zero. Thusif 2|X| >

∏di=1(bi + 1), one can find a proper subset

X of X such that b ∈ int_cone(X). �

Lemma 4. Let X ⊆ Zd be a finite set of integer vec-tors and let b ∈ int_cone(X). If

|X| > d log

(2|X| max

x∈X‖x‖∞ + 1

), (1)

then there exists a proper subset X of X such that b ∈int_cone(X).

Proof. Suppose that b = ∑x∈X �xx with �x > 0 inte-

ger for all x ∈ X. Let n denote the cardinality of X,n=|X|. Suppose that n > d log(2n maxx∈X‖x‖∞+1).For every subset X ⊆ X, ‖∑x∈X x‖∞ is boundedby n maxx∈X‖x‖∞. This implies that the numberof different vectors which are representable as thesum of vectors of a subset X of X is boundedby (2n maxx∈X‖x‖∞ + 1)d . By our assumptionwe have 2n > (2n maxx∈X‖x‖∞ + 1)d . Thereforethere exist two subsets A, B ⊆ X, A �= B, with∑

x∈A x = ∑x∈B x and we can proceed as in the

proof of Lemma 3. �

Proof of Theorem 1. Part (i) follows immediatelyfrom Lemma 3. To prove part (ii), it suffices to showthat |X| > 2d log(4dM) implies (1). Suppose that|X| > 2d log(4dM), that is, M < 2|X|/(2d)/(4d). Then

d log(2|X|M + 1) < d log

( |X|2d

2|X|/(2d) + 1

)

�d log

(2|X|/(2d)

( |X|2d

+ 1

))

= |X|2

+ d log

( |X|2d

+ 1

)

� |X|2

+ d|X|2d

= |X|. �

To prove Theorem 2 we apply the rewriting tech-nique used in the above proof of Theorem 1 togetherwith a technique which was used by Hayes andLarman [4] to prove that the integer hull of knap-sack polytopes has a polynomial number of vertices

in fixed dimension. Recall that our set X ⊆ Zd isclosed under convexity. This implies that if two pointsx1, x2 ∈ X are congruent to each other modulo 2,then 1/2(x1 + x2) is integer and hence also containedin X. We say that b = ∑

x∈X �xx, �x ∈ Z�0, is arepresentation of b of value

∑x∈X �x with potential∑

x∈X �x

∥∥∥(1x

)∥∥∥, where ‖·‖ is the Euclidean norm in

Rd+1.

Proof of Theorem 2. We show that, if there existsa representation of b of value �, then there exists arepresentation with the same value and at most 2d

nonzero coefficients. Let b = ∑x∈X �xx be the rep-

resentation of b of value � with the smallest poten-tial. If the number of nonzero coefficients is greaterthan 2d , then there exists x1 �= x2 ∈ X such that�x1 > 0,�x2 > 0 and x1 ≡ x2(mod 2). Since X is closedunder convex combinations, 1/2(x1+x2) belongs to X.Suppose without loss of generality that �x1 ��x2 . Then�x1x1 + �x2x2 = (�x1 − �x2)x1 + 2�x2(1/2(x1 + x2)).

Since(

1x1

)and

(1x2

)are not co-linear, we have

(�x1 − �x2

) ∥∥∥∥(

1x1

)∥∥∥∥ + 2�x2

∥∥∥∥(

11/2(x1 + x2)

)∥∥∥∥= (

�x1 − �x2

) ∥∥∥∥(

1x1

)∥∥∥∥ + �x2

∥∥∥∥(

1x1

)+

(1x2

)∥∥∥∥<

(�x1−�x2

) ∥∥∥∥(

1x1

)∥∥∥∥+�x2

(∥∥∥∥(

1x1

)∥∥∥∥+∥∥∥∥(

1x2

)∥∥∥∥)

= �x1

∥∥∥∥(

1x1

)∥∥∥∥ + �x2

∥∥∥∥(

1x2

)∥∥∥∥ .

Thus replacing �1/2(x1+x2) by �1/2(x1+x2) + 2�x2 , �x1

by �x1 − �x2 and �x2 by 0 yields a representation ofb with the same value and smaller potential, which isa contradiction. Therefore � has at most 2d nonzerocomponents. �

3. Applications

Integer programming

We can now prove a corollary concerning integerprograms with equations and nonnegativity constraintson the variables. The result states that, if there existsan optimal solution to such an integer program, thenthere exists one which is polynomial in the number of

F. Eisenbrand, G. Shmonin / Operations Research Letters 34 (2006) 564–568 567

equations and the maximum binary encoding lengthof an integer in the objective function vector and theconstraint matrix.

Corollary 5. Let min{cTy |Ay = b, y�0, y integer

}be an integer program, where A ∈ Zd×n and c ∈Zn. If this integer program has a finite optimum withoptimal value �, then there exists an optimal solutiony∗ ∈ Zm

�0 which satisfies the following:

(i) The number of nonzero components of y∗ is atmost size(b)+ size(�), if A and c are nonnegative.

(ii) The number of nonzero components of y∗ is atmost 2(d + 1)(log(d + 1) + s + 2), where s is thelargest size of a coefficient of A and c.

Proof. The integer vector( �

b

)is an integer conic com-

bination of the column vectors of the matrix(

cT

A

).

The optimal solutions correspond to the coefficients inthe integer conic combinations for

( �b

). The assertion

follows thus from Theorem 1. �

3.1. Cutting stock

Let a, b ∈ Zd>0 and � ∈ Z>0. The cutting stock

problem defined by a, b and � is the integer program

min 1T�,

s.t M� = b,

��0 integer,(2)

where the columns of the matrix M are exactly the inte-ger solutions to the knapsack constraint aTx��, x�0,called patterns. The above integer program was in-troduced by Gilmore and Gomory [3]. The cuttingstock problem is NP-hard, however, it was not known,whether the problem has an optimal solution whoseencoding length is polynomial in the input, see forexample [5,6].

A polynomial algorithm for the case d=2 was givenby McCormick et al. [6]. The authors prove that thereexists an optimal solution in this case, which has atmost three nonzero entries in �. It is open, whetherthe cutting stock problem can be solved in polynomialtime, if the number of sizes d is fixed.

It is easy to see that the cutting stock problem de-fined by a, b and � has a feasible solution if and only ifai �� for all i =1, . . . , d. In this case, the unit vectorsei , i = 1, . . . , d, are among the columns of matrix M

in (2). Using these vectors only, one can obtain a fea-sible solution � with 1T�=∑d

i=1 bi . Thus the problemis feasible and bounded and therefore has an optimalsolution.

Corollary 6. Let a ∈ Zd>0, � ∈ Z>0 and b ∈ Zd

>0define a cutting stock problem (2) and let ai ��, i =1, . . . , d. Then there exists an optimal solution to (2)with at most min

{2size(b), 2d

}nonzero components.

Proof. As mentioned above, ai ��, i = 1, . . . , d, im-plies that an optimal solution exists. Let � denote theoptimal value of (2). Then ��

∑di=1 bi , and therefore

log(�+1)�∑d

i=1 log(bi +1)= size(b). By Corollary5 (i), there exists an optimal solution � such that thenumber of nonzero components is at most size(�) +size(b)�2 size(b). The second bound follows imme-diately from Theorem 2, since the set of patterns isclosed under convex combinations. �

Acknowledgements

We are grateful to András Sebo for many usefulsuggestions, which led to an improvement of the re-sults stated here compared to an earlier version of thispaper. We also thank Tom McCormick for many in-teresting discussions on cutting stock.

This work was partly supported by the German Re-search Council (DFG) as part of the TransregionalCollaborative Research Center “Automatic Verifica-tion and Analysis of Complex Systems” (SFB/TR 14AVACS).

References

[1] W. Bruns, J. Gubeladze, M. Henk, A. Martin, R. Weismantel,A counterexample to an integer analogue of Carathéodory’stheorem, J. Reine Angew. Math. 510 (1999) 179–185.

[2] W. Cook, J. Fonlupt, A. Schrijver, An integer analogue ofCarathéodory’s theorem, J. Combin. Theory Ser. B 40 (1)(1986) 63–70.

[3] P.C. Gilmore, R.E. Gomory, A linear programming approachto the cutting-stock problem, Oper. Res. 9 (1961) 849–859.

[4] A.C. Hayes, D.G. Larman, The vertices of the knapsackpolytope, Discrete Appl. Math. 6 (1983) 135–138.

[5] O. Marcotte, An instance of the cutting stock problem forwhich the rounding property does not hold, Oper. Res. Lett.4 (5) (1986) 239–243.

568 F. Eisenbrand, G. Shmonin / Operations Research Letters 34 (2006) 564–568

[6] S.T. McCormick, S.R. Smallwood, F.C.R. Spieksma, Apolynomial algorithm for multiprocessor scheduling with twojob lengths, Math. Oper. Res. 26 (1) (2001) 31–49.

[7] A. Schrijver, Theory of Linear and Integer Programming,Wiley, New York, 1986.

[8] A. Sebo, Hilbert bases Caratheodory’s theorem andcombinatorial optimization, in: R. Kannan, W.R. Pulleyblank

(Eds.), Proceedings of the First Integer Programmingand Combinatorial Optimization Conference, ON, Canada,University of Waterloo Press, Waterloo, May 28–30, 1990,pp. 431–456.