Expected Frobenius numbers

Journal of Combinatorial Theory, Series A 118 (2011) 525–531

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Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta

Expected Frobenius numbers Iskander Aliev a , Martin Henk b , Aicke Hinrichs c a

School of Mathematics and Wales Institute of Mathematical and Computational Sciences, Cardiff University, Senghennydd Road, Cardiff, Wales, UK b Institut für Algebra und Geometrie, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, D-39106-Magdeburg, Germany c Friedrich Schiller Universität Jena, Fakultät für Mathematik und Informatik, Ernst-Abbe-Platz 2, D-07743 Jena, Germany

a r t i c l e

i n f o

Article history: Received 14 October 2009 Available online 8 January 2010 Keywords: Frobenius number Geometry of numbers Reverse AGM inequality Knapsack polytope

a b s t r a c t Given a primitive positive integer vector a, the Frobenius number F(a) is the largest integer that cannot be represented as a nonnegative integral combination of the coordinates of a. We show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bound. © 2009 Elsevier Inc. All rights reserved.

1. Introduction Let a be a positive integral n-dimensional primitive vector, i.e., a = (a1 , . . . , an ) ∈ Zn>0 with gcd(a) := gcd(a1 , . . . , an ) = 1. The Frobenius number of a, denoted by F(a), is the largest number that cannot be represented as a non-negative integral combination of the ai ’s, i.e.,





n F(a) = max b ∈ Z: b = a, z for all z ∈ Z0 ,

where ·,· denotes the standard inner product on Rn . In other words, F(a) is the maximal right-hand side b, such that the well-known knapsack polytope P (a, b) = {x ∈ Rn0 : a, x = b} does not contain an integral point. From that point of view it is also apparent that the Frobenius number plays an important role in the analysis of integer programming algorithms (see, e.g., [1,17,19,22,25]) and, vice versa, integer programming algorithms are known to be an effective tool for computing the Frobenius number (see, e.g., [9,13,21]). There is a rich literature on Frobenius numbers, and for an impressive survey on the history and the different aspects of the problem we refer to the book [2]. Here we just want to mention that only for n = 2 an explicit formula is known, which was most likely known to Sylvester:

E-mail addresses: [email protected] (I. Aliev), [email protected] (M. Henk), [email protected] (A. Hinrichs). 0097-3165/$ – see front matter doi:10.1016/j.jcta.2009.12.012

© 2009 Elsevier Inc.

All rights reserved.

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I. Aliev et al. / Journal of Combinatorial Theory, Series A 118 (2011) 525–531

F(a) = a1 a2 − (a1 + a2 ). There is a huge variety of upper bounds on F(a). They all share the property that in the worst case they are of quadratic order with respect to the maximum norm of a, say, which will be denoted ˝ and Graby |a|∞ . For instance, assuming a1  a2  · · ·  an , a classical upper bound due to Erdos ham [14] says



F(a)  2an

a1



n

− a1 ,

and, in a recent paper, Fukshansky and Robins [15, Equation (29)] gave an upper bound which is also symmetric in the ai ’s



F(a) 

n (n − 1)2 /Γ ( n2 + 1) 

π n/2

 ai



|a|2

2



− a2i

+1 ,

(1.1)

i =1

where | · |2 denotes the Euclidean norm. The worst case in all the known upper bounds is achieved when the ai ’s are approximately of the same size, and it is also known that in these cases the quadratic order of an upper bound cannot be lowered (see [7,14,23]). On the other hand, Aliev and Gruber [3] recently found an optimal lower bound for the Frobenius number which implies that 1

1

F(a) > (n − 1)! n−1 (a1 a2 · . . . · an ) n−1 − (a1 + · · · + an ).

(1.2) 1+1/(n−1)

Hence, if all the ai ’s are of the same size then the lower bound is only of order |a|∞ . In fact, taking the quotient of the (symmetric) upper bound (1.1) with (1.2), we see that there is always a gap 1−1/(n−1) of order |a|∞ . Thus the next natural and important question is to get information on the Frobenius number of a “typical” vector a. This problem appears to be hard, and to the best of our knowledge it has firstly been systematically investigated by V.I. Arnold, see, e.g., [5,7,8]. In particular, he conjectured that 1+1/(n−1) for a “typical” vector a with 1-norm |a|1 = T [5], and in [6, 2003-5] he F(a) grows like T conjectures that the “average behavior” (for details, see below) is 1

1

F(a) ∼ (n − 1)! n−1 (a1 a2 · . . . · an ) n−1 ,

(1.3)

i.e., it is essentially the lower bound. A similar conjecture for the 3-dimensional case was proposed by Davison [12] and recently proved by Shur, Sinai and Ustinov [24]. Extensive computations support conjecture (1.3) (see [9]). In [10], Bourgain and Sinai proved a statement in the spirit of these conjectures, which says, roughly speaking, that





Prob F(a)/ T 1+1/(n−1)  D   ( D ), where Prob(·) is meant with respect to the uniform distribution among all points in the set





G( T ) = a ∈ Zn>0 : gcd(a) = 1, |a|∞  T . Here the number  ( D ) does not depend on T and tends to zero as D approaches infinity. The paper [4] gives more precise information about the order of decay of the function  ( D ). Their main result [4, Theorem 1.1] implies that



1+1/(n−1)

Prob F(a)/|a|∞

 D n D −2 ,

(1.4)

where n denotes the Vinogradov symbol with the constant depending on n only. In particular, from that result the authors get a statement about the average Frobenius number, namely [4, Corollary 1.2],

sup T

1+1/(n−1) a∈G( T ) F(a)/|a|∞

#G( T )

n 1.

I. Aliev et al. / Journal of Combinatorial Theory, Series A 118 (2011) 525–531

527 1+1/(n−1)

So the order of the average Frobenius number is (essentially) not bigger than |a|∞ close to the sharp lower bound (1.2), but there is still a gap. The main purpose of this note is to fill that gap. We will show

, which is

Theorem 1. Let n  3. Then





1

n −1

Prob F(a)/(a1 a2 · . . . · an ) n−1  D n D −2 n+1 . From this result we will derive the desired statement. Corollary 1. Let n  3. Then

sup

a∈G( T ) F(a)/(a1 a2

1

· . . . · an ) n−1

#G( T )

T

 n 1.

These statements supplement also perfectly recent results on the limit distribution of Frobenius numbers due to Shur, Sinai and Ustinov [24] and Marklof [20]. For instance, in our special setting, [20, Theorem 1] says that



1



lim Prob F(a)/(a1 a2 · . . . · an ) n−1  D = Ψ ( D ),

T →∞

where Ψ : R0 → R0 is a non-increasing function with Ψ (0) = 1. The proof of Theorem 1 is based on a discrete inverse arithmetic–geometric mean inequality which might be of some interests in its own. It will be stated and proved in Section 2. Finally, Section 3 contains the proofs of Theorem 1 and Corollary 1. 2. Reverse discrete AGM inequality

n

n

For x ∈ Rn0 the arithmetic–geometric mean (AGM) inequality states that ( i =1 xi )1/n  n1 i =1 xi . It is known that the “reverse” AGM inequality holds with high probability. More precisely, Gluskin and Milman [16] have shown that

 Prob S n−1

 n 1

 2 1 i =1 x i n

n = > α  cn α −n/2 , √

n ( i =1 |xi |)1/n n( i =1 |xi |)1/n

where c is an absolute constant and Prob S n−1 () is meant with to standard rotation invariant n respect 2 measure on the n-dimensional unit sphere S n−1 = {x ∈ Rn : i =1 xi = 1}. Here we show an analogous statement with respect to the primitive lattice points in the set G( T ). With respect to the uniform distribution on G( T ), let L T : G( T ) → R>0 be the random variable defined by 1

n

i =1 a i . 1/n i =1 a i )

n L T (a) = n

(

Theorem 2. Let α > 1 and let k ∈ {1, . . . , n − 1}. Then there exists a constant c (k, n) depending only on k, n, such that

Prob( L T  α )  c (k, n)α −k . By Markov’s inequality (see, e.g., [18, Theorem 5.11]) the theorem is an immediate consequence of the next statement about the expectation E( L kT ) of higher moments of L T .

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Lemma 1. Let k ∈ {1, . . . , n − 1}. Then there exists a constant c (k, n) depending on k, n, such that



E LkT  c (k, n).

Proof. First we note that there is an absolute constant c such that

#G( T )  cT n .

(2.1)

This follows easily from well-known relations between integration and counting primitive lattice points (see e.g. [11, p. 183, (1)]), but in order to keep the paper self-contained we give a short argument here: let n  2 and let





G2 ( T ) = (a, b) ∈ Z21 : 1  a, b  T , gcd(a, b) = 1 . Since G2 ( T ) × {1, . . . , T }n−2 ⊆ G( T ) it suffices to prove the statement for n = 2, i.e., G2 ( T ). There are at most ( T /m)2 pairs (a, b) ∈ {1, . . . , T } with gcd(a, b) = m. Thus



#G2 ( T )  T

2

1−



∞ 

m

−2

= 2 − π 2 /6 T 2 ,

m =2

which gives (2.1). Now we have



E LkT = = =



=



1 #G( T )

L T (a)k

a∈G( T )

 

1 #G( T )

a∈G( T )

1 n

n

#G( T )

a∈G( T )



nk

i 1 +i 2 +···+in =k



1 #G( T )

i 1 +i 2 +···+in =k



1 #G( T )

k

1 ( ni=1 ai ) n

 1

1

i =1 a i

i 1 +i 2 +···+in =k

1







k i 1 , i 2 , . . . , in

k



nk i 1 , i 2 , . . . , in 1



i

k

i

i

a11 a22 · . . . · ann

k ( ni=1 ai ) n

T 

n 

i j −k/n

aj

a1 ,a2 ,...,an =1 j =1

 T n 

nk i 1 , i 2 , . . . , in

i j −k/n

aj

(2.2)

.

j =1 a j =1

T

−k/n is bounded from above by c¯ (k, n) T 1−k/n , where c¯ (k, n) is a Since for k < n the sum m=1 m constant depending only on k and n, we find T 

i j −k/n

aj

 c¯ (k, n) T 1+i j −k/n .

(2.3)

a j =1

Thus, for i 1 + i 2 + · · · + in = k we obtain T n  

i j −k/n

aj

 c¯ (k, n)n T n .

j =1 a j =1

Hence we can continue (2.2) by



E LkT 

1 #G( T )

 i 1 +i 2 +···+in =k

1 nk

Finally, with (2.1) we get the assertion.



k i 1 , i 2 , . . . , in

2

 c¯ (k, n)n T n = c¯ (k, n)n

Tn #G( T )

.

I. Aliev et al. / Journal of Combinatorial Theory, Series A 118 (2011) 525–531

529

Remark 1. With a little more work one can prove Lemma 1 for any positive number k < n, not only for integers. Next we want to point out that the arguments in the proof of Lemma 1 lead to the following lower bound on the random variable X T : G( T ) → R>0 given by

X T (a) = n

F(a)

i =1 a i )

(

1 n −1

.

Proposition 1. Let n  2. Then there exists a constant depending only on n such that 1 1

E( X T )  (n − 1)! n−1 1 − c (n) T − n−1 .

Proof. On account of the lower bound (1.2) on F(a) it remains to show that

n



1 #G( T )

ai

1

i =1  c (n) T − n−1 .

n 1 n −1 ( a ) a∈G( T ) i =1 i

Following the argumentation in (2.2) we find

n



i =1 a i

n 1 n −1 a∈G( T ) ( i =1 ai )

T 





n 

1 i j − n− 1

aj

i 1 +i 2 +···+in =1 a1 ,a2 ,...,an =1 j =1 T 

=n

a1 ,a2 ,...,an =1

=n

T 

1− 1 a 1 n −1

1 1− n − − 1 1 a 2 n −1

a1



a 1 =1

T 

1 − n− 1

1 − n− 1

· . . . · an n−1

a2

.

a 2 =1

Thus, analogously to (2.3), and on account of (2.1) we obtain

n



1 #G( T )

1 1 n−1 1 1 i =1 a i  c˜ (n) T 2 − n −1 T 1 − n −1  c (n) T − n−1 .

n 1 #G ( T ) n −1 a∈G( T ) ( i =1 ai )

2

3. Proof of Theorem 1 and Corollary 1 We keep the notation of the previous section. First we note that (1.4) is certainly also true for any other norm in the denominator, in particular for the 1-norm | · |1 . Thus



Prob

F(a) 1+ 1 |a|1 n−1



 D n D −2 .

(3.1)

Secondly, we observe that

 Prob



1 1+ n − 1

|a|1

(a1 · . . . · an )

1 n −1

γ

 = Prob

n

( n1 |a|1 ) n−1 1 n −1



  n−n 1  1

n

γ

(a1 · . . . · an )  n   n−n 1  1 = Prob L Tn−1  γ n   (n−1)2 1 n −1 = Prob L T  γ n n γ − n , n

(3.2)

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I. Aliev et al. / Journal of Combinatorial Theory, Series A 118 (2011) 525–531

where in the last inequality we have applied Theorem 2 with k = n − 1. Together with (3.1) we obtain







Prob X T (a)  β = Prob

1 1+ n − 1

·

|a|1 

|a|1

1

(a1 · . . . · an ) n−1 

F(a)

 Prob

1 1+ n − 1

β

t

 + Prob

|a|1

n β −2t + β −



1 1+ n − 1

F(a)

(n−1)2 n

(1−t )

β 1 1+ n − 1

|a|1

1

(a1 · . . . · an ) n−1

 β 1−t



,

for any t ∈ (0, 1). With t = (n − 1)/(n + 1) we finally get





n −1

Prob X T (a)  β n β −2 n+1 , which shows Theorem 1. For the proof of Corollary 1 we note that

∞ E( X T ) =

∞ Prob( X T > x) dx  1 +

0

∞ Prob( X T > x) dx n 1 +

1

n −1

x−2 n+1 dx.

1

For n  4, the last integral is finite and so we have

E( X T ) n 1.

(3.3)

For the case n = 3 we just note that on account of Remark 1 one can also bound Prob( X T (a)  β) by a function like β −(1+ ) and so we also get (3.3) in this case. Together with Proposition 1, Corollary 1 is proven. Acknowledgments We thank Professor Schinzel and the anonymous referees for valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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