Irreducible decomposition of monomial ideals has an increasing number of
applications from biology to pure math. This paper presents the Slice Algorithm
for computing irreducible decompositions, Alexander duals and socles of
monomial ideals. The paper includes experiments showing good performance in
practice.Comment: 25 pages, 8 figures. See http://www.broune.com/ for the data use

Bayer

Bigatti

Bjarke Hammersholt Roune

Block

Cormen

Galler

Hoşten

Jarrah

Miller

Roune

Sturmfels

Sullivant

English

arXiv

A Gröbner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible decompositions and Hilbert functions are also presented

Bjarke Hammersholt  Roune

CiteSeerX

Bjarke Hammersholt Roune
Department of Computer Science
University of Aarhus
IT-parken, Aabogade 34
DK-8200 Aarhus N, Denmark
Solving Thousand Digit Frobenius Problems
Using Gröbner Bases
Published in Journal of Symbolic Computation
volume 43 issue 1, January 2008
doi:10.1016/j.jsc.2007.06.002
Abstract
A Gröbner basis-based algorithm for solving the Frobenius Instance Problem is presented, and
this leads to an algorithm for solving the Frobenius Problem that can handle numbers with
thousands of digits. Connections to irreducible decompositions and Hilbert functions are also
presented.
1. Introduction
Let p1, . . . , pn be relatively prime positive integers and let p = (p1, . . . , pn). An in-
teger is p-representable if it can be written as v · p for some v ∈ Nn. Determining
p-representability is known as the Frobenius Instance Problem, and we present a new
Gröbner basis-based algorithm that solves it. 1 Our algorithm differs from the classical
way to solve integer programs using Gröbner bases due to Conti and Traverso (1991)
(see also Pottier (1994)) by not adding any auxiliary variables to the problem.
The Frobenius Number f∗p is the largest integer that is not p-representable, and such
an integer exists by Proposition 2 below. E.g. if p = (6, 10, 15) then f∗p = 29.
The Frobenius Problem is to compute the Frobenius Number f∗p . A recent algorithm
due to Einstein et al. (2007) can solve Frobenius problems even if the pi have thousands
of decimal digits. Here we describe a simpler variant of that algorithm that performs
better.
URL: http://www.broune.com/ (Bjarke Hammersholt Roune).
1 Malkin (2006) has discovered this independently.
Preprint submitted to Elsevier Science 9 January 2008
The algorithm first computes a Gröbner basis and then determines the Frobenius num-
ber based on that. Einstein et al. (2007) use a novel algorithm based on the Fundamental
Domain that in effect computes this Gröbner basis, and they report that this is much
faster than using Buchberger’s algorithm in their implementation.
However, the benchmarks in section 7 show that the program 4ti2 [4ti2 team (2006)],
which implements Buchberger’s algorithm in a special case, outperforms the Fundamental
Domain-based implementation from Einstein et al. (2007). The record for random pi with
11 decimal digits was n = 11, but we can now reach n = 13. Performance is also improved
on smaller examples.
We show that the second step of the algorithm of Einstein et al. (2007) can be
rephrased as the computation of the irreducible decomposition of a certain monomial
ideal. The final step of the algorithm is to maximize a linear function over the decom-
position, and we show that this essentially computes the index of regularity of the ideal.
We define these terms when they are needed.
See Jensen et al. (2007) for a geometric version of some of the ideas in this paper in
terms of maximal lattice free bodies. That paper is joint work with Niels Lauritzen and
Anders Nedergaard Jensen.
We refer to the book by Ramı́rez Alfonśın (2005) for more background on the Frobenius
Problem. We wish to thank Anders Nedergaard Jensen, Niels Lauritzen, Daniel Lichtblau
and Stan Wagon for helpful discussions about Frobenius numbers.
2. Preliminaries
Let ei ∈ Nn, i = 1, . . . , n, be the vector whose entries are all zero except that there is
a 1 at position i. The p-degree of a vector v ∈ Zn is v · p. If v ∈ Zn then define v+ ∈ Nn
as below and define v− := (−v)+.
v+i :=
 vi, for vi ≥ 00, for vi < 0
We will need to consider the lattice ideal IL defined by
IL :=
〈
xv
+
− xv
−
∣∣∣ v ∈ Zn and v · p = 0〉 .
We will compute a Gröbner basis G of IL. The term order ≤ we will use first considers
the p-degree of the exponent vector of a term and then the reverse lexicographic order
where x1 < x2 < · · · < xn.
Example 1. If p = (2, 3) then 1 = x(0,0) ≤ x(1,0) ≤ x(3,0) ≤ x(0,2) ≤ x(4,5) = x41x52.
We refer to Cox et al. (1997) for more details.
Proposition 2. Only finitely many integers t ∈ N are not p-representable.
Proof. As p1, . . . , pn are relatively prime, iterated use of the Extended Euclidean Al-
gorithm provides us with a vector v ∈ Zn such that v · p = 1. Let m := minni=1 vi and
define u := v + p1|m|(1, . . . , 1). Then u + iv ∈ Nn for i = 0, . . . , p1 − 1 and therefore
(u+ iv) · p = u · p+ i are p1 consecutive p-representable numbers. 2
2
3. An Algorithm That Solves The Frobenius Instance Problem
We claim that the following algorithm determines if t ∈ N is p-representable.
Step 1: Compute an a ∈ Zn such that a · p = t, a1 ≤ 0 and ai ≥ 0 for i = 2, . . . , n.
Step 2: Divide xa
+ − xa− by G giving remainder xw(xc+ − xc−) for some w ∈ Nn where
xc
− ≤ xc+ .
Step 3: Then t is p-representable if and only if c ∈ Nn.
Step 1 We first find an a ∈ Zn such that a · p = t. One way to do this is to use the
Extended Euclidean Algorithm iteratively to find a b ∈ Zn such that b · p = 1 and
then let a := tb. As (−
∑n
i=2 pi, p1, . . . , p1) has p-degree zero and the sign pattern
(−,+, · · · ,+), we can assume that a also has this sign pattern by adding a sufficiently
large multiple of this vector to a.
Step 2 This step requires knowing the Gröbner basis G, and computing G is the most
time consuming part of the algorithm. Once G has been computed the polynomial
division itself is comparatively fast.
Step 3 Observe that the division algorithm ensures that c will have no more negative
entries than a does, so c is negative at most in the first coordinate. As xc
+
is not
reducible by G, the following lemma tells us that if t is p-representable then c1 ≥ 0 so
that c is a p-representation of t since c · p = a · p = t.
Lemma 3. Let a ∈ Zn such that ai ≥ 0 for i = 2, . . . , n and a1 < 0. If a · p is p-
representable then there exists a g ∈ G such that in≤(g)|xa
+
.
Proof. As a · p is p-representable there exists a b ∈ Nn such that a · p = b · p. Letting
d := a − b this implies that d · p = 0 whereby h := xd+ − xd− ∈ IL. Thus there exists a
g ∈ G such that in≤(g)|in≤(h). We will prove that in≤(g)|in≤(h) = xd
+ |xa+ .
in≤(h) = xd
+
: d+ and d− have the same p-degree and d1 = a1 − b1 < 0.
xd
+ |xa+ : This follows from b ∈ Nn. 2
Note that ≤ can be replaced with any term order that first considers the p-degree of
the exponent vector and then the reverse lexicographic order on the first variable.
4. An Algorithm That Solves The Frobenius Problem
The general idea of the algorithm is that we can represent f∗p by a certain vector that
has p-degree f∗p , and that this vector has certain properties (see Proposition 4). It turns
out that only finitely many vectors have these properties, so we can look through all of
them, and then the one with maximal p-degree will be the vector that represents f∗p , i.e.
it will have p-degree equal to f∗p (see Proposition 5).
Proposition 4 spells out the properties mentioned above.
Proposition 4. Let c ∈ Zn be the vector resulting from running the algorithm from
Section 3 on t = f∗p . Then the following holds.
(M1) c1 = −1 and ci ≥ 0 for i = 2, . . . , n.
(M2) xc
+
cannot be reduced by any element of G.
3
(M3) x(c+ei)
+
can be reduced by some gi ∈ G for i = 2, . . . , n.
Proof. (M1): It holds by construction that c1 < 0 and that ci ≥ 0 for i = 2, . . . , n.
Let c′ := c+ e1. Then x(c
′)+ = xc
+
is not reducible by G and c′ · p is p-representable as
c′ · p > f∗p . Then Lemma 3 implies that c′1 ≥ 0 whereby c1 = −1.
(M2): This holds by construction.
(M3): We see that (c+ ei) · p is p-representable as it is strictly larger than f∗p . Thus
Lemma 3 provides a gi ∈ G such that in≤(gi)|x(c+ei)
+
. 2
Let Mp be the set of vectors c ∈ Zn that have the properties (M1), (M2) and (M3)
from Proposition 4. The idea is to compute Mp and then use Proposition 5 below to find
f∗p .
Proposition 5. The following holds.
(i) Mp is finite.
(ii) If a ∈Mp then a · p is not p-representable.
(iii) f∗p = max {a · p|a ∈Mp}
Proof. (i): Let a ∈ Mp and i ∈ {2, . . . , n}. Let g ∈ G such that in≤(g)|x(a+ei)
+
. As
in≤(g) does not divide xa
+
we can infer that ai + 1 is the exponent of xi in in≤(g). Thus
there are at most |G| possibilities for what ai can be.
(ii): Lemma 3 shows that this follows from (M1) and (M2).
(iii): Proposition 4 shows that there is an a ∈Mp such that a · p = f∗p . This and part
(ii) above shows what we need. 2
We can compute Mp as follows. We saw in the proof of Proposition 5 that if i ∈
{2, . . . , n} and a ∈Mp then ai + 1 is the exponent of xi in in≤(gi) for some gi ∈ G. Thus
we can run through all possible values of ai for i = 2, . . . , n and only keep those a that
have properties (M1), (M2) and (M3).
This algorithm is easy to understand and implement, but it requires us to look through
up to |G|n−1 possibilities. The External Corner Algorithm from Einstein et al. (2007)
is a more efficient algorithm for computing Mp which usually dramatically reduces the
number of possibilities that need to be examined.
In Section 5 we will define the irreducible decomposition of a monomial ideal, and we
will prove that Mp corresponds to the irreducible decomposition of the initial ideal of
IL. Roune (2007) shows how an algorithm that is very similar to the External Corner
Algorithm can compute irreducible decompositions of monomial ideals in general in much
less time than the best competing programs.
5. A Connection To Monomial Irreducible Decompositions
We need a few more definitions. Let I be a monomial ideal and define the function
φ by φ (v) = 〈xvii |vi > 0〉 for v ∈ Nn except that φ (1) = 〈1〉. An ideal of the form φ (v)
is called irreducible and the irredundant irreducible decomposition of I is the unique
minimal subset D ⊆ Nn such that I =
⋂
v∈D φ (v). Thus the irredundant irreducible
decomposition of I :=
〈
x21, x1x2
〉
is {(1, 0), (2, 1)} as I = 〈x1〉 ∩
〈
x21, x2
〉
.
4
An ideal is artinian if there exists a t ∈ N such that xti ∈ I for i = 1, . . . , n. Note
that in≤(IL) is artinian if we first project out the variable x1, since x
p1
i − x
pi
1 ∈ IL and
therefore in≤(x
p1
i − x
pi
1 ) = x
p1
i ∈ in≤(IL) for i = 2, . . . , n.
We claimed in Section 4 that Mp corresponds to the irreducible decomposition of the
initial ideal of IL, and Proposition 6 proves this claim.
Proposition 6. The set M ′ := {(a1+1, . . . , an+1)|a ∈Mp} is the irreducible irredundant
decomposition D of in≤(IL).
Proof. Let a ∈ Zn and let a′ := a−
∑n
i=1 ei. Consider the following statements.
(1) a ∈ D
(2) a1 = 0, ai ≥ 1, in≤(IL) ⊆ φ (a) and in≤(IL) 6⊆ φ (a+ ei) for i = 2, . . . , n.
(3) a′1 = −1, a′i ≥ 0, x(a
′)+ /∈ in≤(IL) and x(a
′+ei)
+ ∈ in≤(IL) for i = 2, . . . , n.
(4) a′ ∈Mp
We will prove that these statements are equivalent.
(1)⇔ (2): The initial ideal of IL contains no monomial that is divisible by x1, and
this implies that a1 = 0 for a ∈ D. Now that we have handled the first entry, we can
project out x1 and thereby get an artinian ideal.
When working with artinian ideals the elements of the decomposition consists of vec-
tors without zero entries, and if a has no zero entries, then φ (a+ ei) ( φ (a).
To get the irredundant irreducible decomposition of an ideal we write the ideal as an
intersection of irreducible ideals that are as small as possible, and this is exactly what
(2) expresses since the projected ideal is artinian.
(2)⇔ (3): By Lemma 7 below.
(3)⇔ (4): By definition. 2
Lemma 7. Let m ∈ Nn and let I be a non-zero monomial ideal. Then xm ∈ I if and
only if I 6⊆ φ (m+
∑n
i=1 ei).
Proof. Let a ∈ I be a monomial. Then a|xm if and only if a /∈ φ (m+
∑n
i=1 ei). 2
6. A Connection To The Hilbert Function
The p-weighted Hilbert function HFI : N → N of a monomial ideal I is defined such
that HFI(t) is the number of monomials xm /∈ I where m has p-degree t.
Proposition 8. HFin≤(IL) (t) is equal to 1 if t is p-representable and 0 otherwise.
Proof. HFin≤(IL) (t) ≤ 1: Let m1,m2 be two vectors of p-degree t. Then x
m1−xm2 ∈ IL
whereby the initial term is in in≤(IL). Thus at most one of xm1 and xm2 is not in in≤(IL).
HFin≤(IL) (t) = 1⇒ representability: If HFin≤(IL) (t) = 1 then there is a monomial
xm /∈ in≤(IL) where m has p-degree t. Thus t is p-representable.
representability⇒ HFin≤(IL) (t) = 1: Run the algorithm from Section 3 on t. The
resulting vector c ∈ Nn is such that xc /∈ in≤(IL). 2
5
We can infer that HFin≤(IL)
(
f∗p
)
= 0 and that HFin≤(IL) (t) = 1 for all integers
t > f∗p . The integer at which the Hilbert function becomes equal to a polynomial is
known as the index of regularity, so in this case the index of regularity is f∗p + 1.
The elements of Mp correspond to the maximal monomials outside of in≤(IL) accord-
ing to divisibility (disregarding the first variable), and what the algorithm from Section
4 does is to maximize the dot product with p over Mp. This amounts to maximizing the
dot product over the vectors m such that xm+e1 /∈ in≤(IL) and m1 = −1.
We can multiply any monomial not in in≤(IL) with x1 and get something still not
in in≤(IL), so anything that goes on in the first variable does not prevent the Hilbert
function from becoming a polynomial. Thus the algorithm can be interpreted as finding
the maximal point that prevents the Hilbert function from becoming a polynomial, which
is to say that it computes the index of regularity.
We conclude that the algorithm of Einstein et al. (2007) can be interpreted as com-
puting an index of regularity.
7. Benchmarks
Roune (2006) has written an implementation called Frobby of the algorithm described
in this paper, and here we compare Frobby to the implementation of Einstein et al.
(2007). Figure 1 displays the collected data. “Intractable” means intractable according
to the authors of that software package. Note that the most time consuming part of the
algorithm usually is to compute the Gröbner basis. Frobby and Mathematica are the
only programs that can handle input numbers pi as large as those in figure 1.
Frobby uses the program fplll due to Stehlé (2006) to obtain an LLL-reduced lattice
basis and then computes the Gröbner basis of the lattice ideal IL from that using the
program 4ti2 [4ti2 team (2006)]. Frobby then computes the Frobenius number by com-
puting an irreducible decomposition of in≤(IL) using the algorithm due to Roune (2007),
which is similar to the External Corner Algorithm. Frobby is written in C++ and the
source code is available under the GNU General Public License (GPL).
The implementation of Einstein et al. (2007) is available as a part of Mathematica
and is written mostly in C. The major difference from Frobby is that the Gröbner basis
is computed using a different algorithm than Buchberger’s.
All the inputs were randomly generated using genuinely random radioactive decay via
the service provided by Walker (2006) except the n = 11 input which was provided by
Stan Wagon who pseudo-randomly generated it using Mathematica. We wish to thank
Daniel Lichtblau for carrying out the Mathematica benchmarks.
All the benchmarks were run on machines with a 3.0 GHz Pentium 4 CPU with 1 GB
RAM except the Mathematica benchmark on the n = 11 input which was run on a 3.2
GHz Pentium 4.
References
4ti2 team, 2006. 4ti2 version 1.3 – a software package for algebraic, geometric and com-
binatorial problems on linear spaces. Available at http://www.4ti2.de.
Conti, P., Traverso, C., 1991. Buchberger algorithm and integer programming. In:
AAECC-9: Proceedings of the 9th International Symposium on Applied Algebra, Al-
gebraic Algorithms and Error-Correcting Codes. Springer-Verlag, London, UK, pp.
130–139.
6
n blog10(min pi)c+ 1 |G| Mathematica Frobby
4 10000 7 620s 3.5s
4 800 10 4.0s 0.3s
5 150 34 3.8s 0.3s
6 70 131 5.5s 0.3s
6 80 148 5.6s 0.3s
6 90 112 6.8s 0.3s
6 100 140 9.5s 0.3s
8 30 2099 80.2s 11.8s
11 11 27037 43.3h 0.5h
12 11 56693 intractable 2.5h
13 11 170835 intractable 49.7h
Fig. 1. The benchmark data.
Cox, D., Little, J., O’Shea, D., 1997. Ideals, Varieties, and Algorithms. Springer.
Einstein, D., Lichtblau, D., Strzebonski, A., Wagon, S., 2007. Frobenius numbers by
lattice point enumeration. Integers 7, available at http://www.integers-ejcnt.org/.
Jensen, A. N., Lauritzen, N., Roune, B. H., 2007. Maximal lattice free bodies, Frobenius
numbers and test sets. Eprint arXiv:0705.4439.
Malkin, P. N., 2006. Truncated Markov bases and Gröbner bases for integer programming.
Eprint arXiv:math/0612615v1.
Pottier, L., 1994. Gröbner bases of toric ideals. INRIA Rapport de recherche 2224, avail-
able at http://hal.inria.fr/inria-00074446/en/.
Ramı́rez Alfonśın, J. L., 2005. The Diophantine Frobenius Problem. Oxford University
Press.
Roune, B. H., 2006. Frobby – a software package for computing Frobenius numbers and
irreducible decompositions of monomial ideals. Available at http://www.broune.com/
frobby/.
Roune, B. H., 2007. The label algorithm for irreducible decomposition of monomial ideals.
Eprint arXiv:0705.4483.
Stehlé, D., 2006. fplll version 1.3 – an lll-reduction program. Available at http://perso.
ens-lyon.fr/damien.stehle/english.html.
Walker, J., 2006. Hotbits – genuine random numbers, generated by radioactive decay.
Available at http://www.fourmilab.ch/hotbits.
7


Solving Thousand Digit Frobenius Problems Using Gröbner Bases

AbstractIrreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice

Roune, Bjarke Hammersholt

Elsevier - Publisher Connector 