Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7, D-14195 Berlin
Test sets of the knapsack problem and simultaneous diophantine approximation
Martin Henk and Robert Weismantel
Preprint SC9713 (March 1997)
Test sets of the knapsack problem and simultaneous diophantine approximation Martin Henk*
Robert Weismantel^
Absact This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the corresponding cone the integer analogue of Caratheodory's theorem applies when the numbers are divisible. We show that the elements of the minimal Hubert basis of the dual cone of all minimal integral solutions of a linear diophantine equation yield best approximations of a rational vector "from above". A recursive algorithm for computing this Hilbert basis is discussed. We also outline an algorithm for determining a Hilbert basis of a family of cones associated with the knapsack problem. K e y w o r d s : k n a p s c k problem, simultaneous diophantine approximation, diophantine equation, Hilbert basis, test sets.
Introduction This paper deals with the study of test sets of the knapsack problem and simul taneous diophantine approximation. Both topics play a role in various branches of m a t h e m a t i c s such as number theory, geometry of numbers and integer pro gramming. From the viewpoint of integer p r o g a m m n g , minimal integral solutions of a linear diophantine equation allow to devise a exact primal algorithm for solving knapsack problems in non-negative integer variables, m a x c x : a x = ß, x £ N n ,
(I- 1 )
where c £ 7L, a £ (N \ {0}) and ß £ N . More precisely, the primal methods t h a t we consider here are a u g m e n t t i o n algorithms, and the question we address is to describe the set of all possible a u g m e n t t i o n vectors. This leads us to test sets * S u p p o d by a "Leibniz P r i s " of the Geman Science Foundaion (DFG) awarded to M. Grötschel. ' Supported by a "Gerhard-Hess-Forschungsförderpreis" of the German Science Foundation (DFG)
A est set is a collecion of all augmenting d i r c t i o n s t h a t one needs in orde to guarantee t h a t every non-optimal feasible point of a linear integer program can be improved by one member in the test set. There are various possible ways of defining test sets depending on the view t h a t one takes: the Graver test set is naturally derived from a 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 t h a t is a classical field of algebra, [CT91]. We refrain within this p e r from introducing all these three kinds of test sets, but concentrate on the G a v e r test set, only. In order to introduce the Graver test set for the family of knapsack problems with varying c £ 7Ln and £ N, the notion of a rational polyhedral cone and its Hilbert basis is needed. D e f i n i t i o n 1.1. For z1,
. , zm £ 7n,
the set lJ2X
p o s { z . ,z}=
A e M
>o
is called a rational polyhedral cone. It is called pointed if there exists a hyper plane { i £ l : aTx = 0} such that {0} = {x
x,y)
b
jVj
where we always assume t h a t a = (a n > m > 1 and a ci2 < • • • < a n+
pos {eje
K„
an)T £ H b < • • •< .
(pi, . , bm)T £ N It is easy to see th (2.
: l i < , l j < m }
where e £ M n + m denotes the th unit vector. The m n i m a l Hilbert basis of Kn , is denoted by %n,mne of the major results of this paper is to show t h a t every element in tLn, satisfies n m special inequalities t h a t generalize the two inequalities YH= xi ^ bm nd ^ 7 = 1 Vj ^ an proved by Lambert ([L87]) and i n d e p e n d e t l y by D c o n i s , G r a h a m & Sturmfels [DGS94] T h e o r e m 2 . 1 . Every (x,y) -
[ji
£ ?„
iequalities - &;" Vj, x
bj
„ ft
satisfies the
%
0, and N £ TL, N 0. There exists an eleme . , qn+) of the miimal Hubert basis o/pos { e 1 , ., p} such that = m n+
n-
n+ 0,
n+
(— n
Among all solutios is the uique o
of this diophatine approximatio ith smallest deomiator
^ n + lJ
+
problem
(
q
J i
.
,
g
"
A recursive algorithm for the Hubert basis o C(p) and the knapsack cone We have motivated in the previous sections why the Hilbert basis of the knapsack cone and the cone of best approximations from above is of particular interest. In this section we treat algorithmic questions related to these bases. We first deal with the cone C(p) = pos {e 1 , . . , en ,p} M n + related to the best approximations from above. Applying a unimodul t r a n s f o r m t i o n we may assume t h a p=(p,..,Pn+)eEn+1. We remark t h a t it is trivial to find a Hilbert basis of (p), because it is well known t h a t { e 1 , . . . , en,p} U {z £ Zn+1 : z = J2"iXie + K+iP,0 < A < 1} actually is a Hilbert basis of C(p) (cf. [C31]). All we are left with is to enumerate these integral points. However, in general, the size of this Hilbert basis is exponentially larger t h n the size of the minimal Hilbert basis, and, of course, we are interested in computing a "small" one. We proceed in an inductive fashion to compute the basis of ): Let H e the m n i m a l Hilbert basis of the 2 d i m e n s i o n a l cone C
p o s { ( p
}
n +
It is clear t h a e 1 £ H. Let (w\,hi) < . < (wmh) £ H2 \ {e 1 } be the +1 remaining elements in Hj. It follows t h a t , for every x £ C(p)" with xn+i > 0, the coordinate xn+ has a representation of the form xn+ i l h with D e f i n i t i o n 4 . 1 . For ij
= Ix£
(p)
£ {1,
.,
"+ : 3
.
1}
let
, j £ N, 0
n+
^
hj
I We say that {g1, . . , gf} C L are generators there exis ., £ N such that x =
of a set v
for every x £
Noting ha
m
n+,
the following s t a m e t is i m m d i e .
L e m m a 4 . 2 . The unio of {e 1 defes a Hilbert basts of ()
. ,ep}
ad
a set of geerators
In f c t , an inclusionwise ordering of all the subsets
of
is possible.
L e m m a 4 . 3 . For every i < i' < j < j ' £ { 1 , . . . ,m — 1}, a minimal set of geerators of3 ( r t . iclusion) defes a subset of any set of geerators of C On account of L e m m a 4.2 it sufices to find generators of to determine a Hilbert basis of C) A set of generators of computed in a recursive m n n e r .
l
in order ~l can be 1
l g o r i t h m 4 . 4 . R e c u r s i o n f o r m u l a t o find t h e g e n e r a t o r s o f (1) For i = 1, . . . ,m — 1 determie geerators of C1 l (2) For i = — 2, . , 1 determie geerators of
The reason why this recursion makes sense is t h a the task of nding a set of generators of C" can be solved with a procedure to determine the Hilbert basis of a cone similar to C'(p), but in one dimension less. Secondly, if one has, for i = m 2, . , 1 as input a set of generators of %% and Jt+1,m~1 ^ then x one can devise a procedure t h a t returns generators of . This is the idea behind the recursion. L e m m a 4 . 5 . Let p (pi, .. . ,pn-hipn+i) £ H and let e 1 . , e 1 de note the first n — 1 unit vectors of 1 . For i £ {1, .. . , m — 1}, let Hl be a Hubert basis of the ndimensional co C{p) = p o s j e 1 , . )enl,p). The set {(xi,. . . , x n z z h ) : (x . , x z ) £ FT 0 zh ) is a set of geerators of e m m a 4.5 shows t h a t a set of generators of ' can easily be reconstructed from a Hilbert basis of the cone C'(p). This is where the inductive step comes into play. In order to solve Step (2) of Algorithm 4.4 one must be able to turn a 1 m1 %m l set of generators of C'% and , l n t o a set of generators of C ' ~ . This ask may again be solved with a recursive algorithm t h a t would read as follows. l g o r i t h m 4.6. ( R e c u r s i o n t o
find
C
l
)
Input: A ordered set {g1, . . . , gf} of geerators < : PnU > Pn+iz}- Then e 1 G H. et (h, w\) < . .. < (h, m) be all the elements of H \ {e 1 } ordered in this way. For i G {1, .. ,m — 1} we introduce a parameter A8 to denote the maximal natural number such t h a t A;/ij < / i + i Then we know t h a t for every x G C(p)t> C\7J1"+l with x ^ 0, be written as Xn = ^2 l / v With yL/i, • , m G N . We define, for every i G {1, . , — 1}, the set : {« G C ( p ) £ n n + 1 : A,-+1 > x n Xi + n > 0, j / G Uj=i , A e {0, . . , }}. Realizing t h a a Hilbert basis of C(p)* consists of the union of the element hme — wmen+1, the set {e 1 , . . ,en+1} and a generating set of m_1 C , the following recursive procedure to d e t e r m n e the Hilbert basis of C(p) ecomes obvious. l g o r i t h m 4.8. For i = 1,
.,
1 determie
a set of geerators
of
Finally, we r e m r k t h a t a similar recursion can be formulated to d e t e r m n e the Hilbert basis of a knapsack cone Kn,, see Section 2.
eferences CT91]
P. Conti, C. Traverso, Buchberger algorithm and integer programming, Proceedings A A E C C (New Orleans) Springer LNCS 539, 130 - 139 (1991)
CFS86] W . Cook, J. Fonlupt, and A. Schrijver, A nteger alogue of Caratheodory's theorem, J. Comb. Theory (B) 0, 1986, 63-70. C31]
J.G. van der Corput, Über Systeme von linearhomogenen Gleichu ge nd Ungleichungen, Proceedings Koninklijke kademie van Wetenschappen te m s t e r d a m 34, 368 - 371 (1931).
[DGS94] P. D c o n i s , R. G r a h a m , and B. Sturmfels, Primitive partition iden tities, Paul Erdös is 80, Vol II, Janos Bolyai Society, Budapest, 1 2 0 (1995) [GP79]
F.R. Giles and W . R . Pulleyblank, Total dual integrality polyhedra, Lineare Algebra Appl 25, 191 - 196 (1979).
[G1873] P. G o r d n , Über die Auflösung linearer Gleichuge ficiente, Math. nn. 6, 23 - 28 (1873)
ad
iteger
mit reelle
Coef
[G75]
J. E. Graver, On the foundations of linear and integer programmig M t h e m a t i c a l Programming 8, 207 - 226 (1975).
I
[L87]
J.L. Lambert, line borne pour les generateurs postwes d'une equation diophatienne lieaire, 3 0 5 , Serie I 1987, 39-40.
S86]
H. E. Scarf, Neighborhood systems for productio ities Econometrica 54, 507 - 532 (1986).
S90]
A. Sebö, Hubert bases Caratheodory's Theorem and combiatorial optimizatio, i Proc. of the I P C conference, Waterloo, Canada, 1990, 431-455.
des solutios entieres C.R. Acad. Sei Paris sets
ith
idivisibil
Konrad-ZuseZentrum für Informationstechnik Berlin, Takustr. 7, D-14195 BerlinDahlem,
[email protected],
[email protected].
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