Integer rounding and modified integer rounding for the skiving stock problem

Discrete Optimization 21 (2016) 118–130

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Discrete Optimization www.elsevier.com/locate/disopt

Integer rounding and modified integer rounding for the skiving stock problem J. Martinovic ∗ , G. Scheithauer Institute of Numerical Mathematics, Dresden University of Technology, 01069 Dresden, Germany

highlights • • • • •

For the first time: profound theoretical investigations on the gap of the SSP. Generalization of Zak’s theorem to arbitrary values of m. Proof of the MIRDP for the divisible case of the SSP. Construction of an infinite number of non-equivalent non-IRDP instances. We show how a gap arbitrarily close to 22/21 can be obtained.

article

info

Article history: Received 29 January 2015 Received in revised form 21 June 2016 Accepted 23 June 2016 Available online 20 July 2016 Keywords: Cutting and packing Skiving stock problem (Modified) integer round down property Gap Continuous relaxation

abstract We consider the one-dimensional skiving stock problem which is strongly related to the dual bin packing problem: find the maximum number of items with minimum length L that can be constructed by connecting a given supply of m ∈ N smaller item lengths l1 , . . . , lm with availabilities b1 , . . . , bm . For this optimization problem, we investigate the quality of the continuous relaxation by considering the gap, i.e., the difference between the optimal objective values of the continuous relaxation and the skiving stock problem itself. In a first step, we derive an upper bound for the gap by generalizing a result of E. J. Zak. As a main contribution, we prove the modified integer round-down property of the divisible case. In this context, we also present a construction principle for non-IRDP instances of the divisible case that leads to gaps arbitrarily close to 22/21. © 2016 Elsevier B.V. All rights reserved.

1. Introduction and preliminaries In this paper, we consider the one-dimensional skiving stock problem (SSP) [1,2] which is strongly related to the dual bin packing problem (DBPP) in literature (see e.g. [3,4] or [5]). In the classical formulation, m ∈ N := {1, 2, . . .} different item lengths l1 , . . . , lm with availabilities b1 , . . . , bm are given, the so-called item supply. We aim at maximizing the number of products with minimum length L that can be constructed by connecting the items on hand. ∗ Corresponding author. E-mail addresses: [email protected] (J. Martinovic), [email protected] (G. Scheithauer).

http://dx.doi.org/10.1016/j.disopt.2016.06.004 1572-5286/© 2016 Elsevier B.V. All rights reserved.

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Such computations are of high interest in many real world applications, e.g. industrial production processes (see [2] for an overview) or politico-economic problems (cf. [3,4]). Furthermore, also neighboring tasks, such as dual vector packing problem [6] or the maximum cardinality bin packing problem [7,5], are often associated or even identified with the dual bin packing problem. These formulations are of practical use as well since they are applied in multiprocessor scheduling problems [8] or surgical case plannings [9]. Throughout this paper, we will use the abbreviation E := (m, l, L, b) for an instance of the SSP with ⊤ ⊤ l = (l1 , . . . , lm ) and b = (b1 , . . . , bm ) . Without loss of generality, we assume all input-data to be positive integers with L > l1 > · · · > lm > 0. The classical solution approach is due to [2] and based on the formulation of Gilmore and Gomory in the context of one-dimensional cutting [10]. Any feasible arrangement of items leading to a final product of minimum length L is called (packing) pattern of E. We always represent a pattern by a nonnegative ⊤ vector a = (a1 , . . . , am ) ∈ Zm + where ai ∈ Z+ denotes the number of items of type i ∈ I := {1, . . . , m} being contained in the considered pattern. For a given instance E, the set of all patterns is defined by  ⊤    PE := a ∈ Zm + l a ≥ L . In [11], the authors slightly improve Zak’s formulation by only considering so-called minimal patterns obtaining a finite model, hereinafter referred to as the standard model, of the skiving stock problem. A pattern a ∈ PE is called minimal if there exists no pattern  a ∈ PE such that  a ̸= a and  a ≤ a hold (componentwise). The set of all minimal patterns is denoted by PE⋆ . Let xj ∈ Z+ ⊤ denote the number how often the minimal pattern aj = (a1j , . . . , amj ) ∈ Zm + (j ∈ J) of E is used, where J = {1, . . . , n} represents an index set of all minimal patterns. Then the skiving stock problem can be formulated as         z ⋆ (E) = max xj  aij xj ≤ bi , i ∈ I, xj ∈ Z+ , j ∈ J .    j∈J j∈J A common (approximate) solution approach consists in considering the continuous relaxation         zc⋆ (E) = max xj  aij xj ≤ bi , i ∈ I, xj ≥ 0, j ∈ J    j∈J j∈J and the application of appropriate heuristics. Practical experience and computational simulations, cf. [2], have shown that there is only a small gap ∆(E) := zc⋆ (E) − z ⋆ (E) for any instance E. Based on the contributions of Baum and Trotter [12] for general linear maximization problems, these observations have initiated the following definitions. A set P of instances has the integer round-down property (IRDP) if ∆(E) < 1 holds for all E ∈ P. An instance E with ∆(E) ≥ 1 is called non-IRDP instance. Furthermore, a set P of instances has the modified integer round-down property (MIRDP) if ∆(E) < 2 is true for all E ∈ P. It is conjectured in [2] that the one-dimensional skiving stock problem possesses the MIRDP. In this paper, we investigate the gap of the skiving stock problem from a theoretical point of view. To this end, we first take the result of [2, Theorem 4], i.e., the proof of the IRDP for all instances with m = 2 item lengths, as an initial point and generalize this inequality to arbitrary values of m. This case and other special cases have also been studied by Marcotte [13] in the context of one-dimensional cutting. In Section 3, we focus intensively on the divisible case of the skiving stock problem where li |L (i.e. there is some γi ∈ N with li /L = γi ) holds for each item length li (i ∈ I). For these instances, we prove the MIRDP by means of the first fit decreasing (FFD) heuristic presented in [3] for the dual bin packing problem. Afterwards, we introduce a construction principle for an infinite number of non-equivalent nonIRDP instances of the divisible case. In a final step, we investigate the gap that can be obtained on the basis of these instances, give some conclusions, and provide an outlook of future research.

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2. A generalization of Zak’s theorem To our best knowledge, there is only few work concerning theoretical aspects of the skiving stock problem’s gap. One of the only results being known in literature is Theorem 4 in [2] where the IRDP for instances with m = 2 item lengths is proved. In the following theorem, we generalize this statement such that it provides an upper bound for the gap of arbitrary instances E = (m, l, L, b). Theorem 1. Let m ∈ N with m ≥ 2 be given and let E = (m, l, L, b) be an arbitrary instance with m types of item lengths. Then ∆(E) < m − 1 holds. Proof. Let n := |J| denote the number of all minimal patterns of E. Then, consider the optimization problem   c⊤ x → max s.t. A I x = b, x ≥ 0 (1) is defined by cj = 1 for j ∈ {1, . . . , n} and cj = 0 for j ≥ n + 1. Note that the where c ∈ Zn+m + columns of A consist of the minimal patterns of E, and I represents the appropriate  identity  matrix. Let j1 , . . . , jm ∈ {1, . . . , n + m} denote the (pairwise different) indices of those columns of A I that belong to the basis matrix of an arbitrary but fixed solution xLP ∈ Rn+m of (1). Thus, we have xLP = 0 for all + j j ∈ {1, . . . , n + m} \ {j1 , . . . , jm }. Case 1: There is an index jk ∈ {j1 , . . . , jm } with jk ≥ n + 1. In this case, we obtain an integer solution by means of   xLP for j ∈ {1, . . . , n}, j xj := 0 otherwise.   Due to xLP = xLP = 0 for all j ∈ {1, . . . , n + m} \ {j1 , . . . , jm }, this immediately implies j j zc⋆ (E) − z ⋆ (E) ≤

n 

xLP − j

j=1

n 

xj =

j=1

(2)

n  

  xLP − xLP j j    j=1 <1

< |{j1 , . . . , jm } \ {jk }| = m − 1. Case 2: jk ≤ n holds for all jk ∈ {j1 , . . . , jm }. In this case, the equality n 

aij xLP = bi j

j=1

holds for all i ∈ I, and we have to consider two cases. Case 2.1: Let n  

  xLP − xLP < 1. j j

j=1

Then, a feasible solution for the integer problem is given by (2), and we obtain zc⋆ (E) − z ⋆ (E) ≤

n  j=1

due to the assumption of this case.

xLP − j

n   j=1

 xLP <1≤m−1 j

(3)

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Case 2.2: Let n  

  xLP − xLP ≥ 1. j j

j=1

   n Defining ∆xj := xLP − xLP ≥ 0 for all j ∈ {1, . . . , n} and a⋆i := bi − j=1 aij xLP for all i ∈ I, we j j j  n obtain, by means of (3), a⋆i = j=1 aij ∆xj for all i ∈ I. Then, a⋆ := (a⋆1 , . . . , a⋆m )⊤ ∈ Zm + describes a packing pattern since we obtain 

l⊤ a⋆ =

m 

li a⋆i =

i=1

≥

n 

m n n m n        li aij ∆xj = ∆xj li aij = ∆xj · l⊤ aj i=1

j=1 n 

∆xj · L = L ·

j=1

j=1

i=1

j=1

∆xj ≥ L

j=1

with the help of the assumption of this case. Without loss of generality, we assume a⋆ to be a minimal pattern. Otherwise, some entries can be repeatedly reduced until this situation is achieved. Thus, a⋆ has to be a column of A, say a⋆ = ak for some k ∈ {1, . . . , n}. Hence, we obtain a feasible solution for the integer problem by slightly modifying (2) to  LP  for j ∈ {1, . . . , n} \ {k},   xj  LP  xj := xj + 1 for j = k,   0 otherwise, since 

A

I



=

i

n 

n    LP      aij xLP + a · x + 1 = aij xLP + aik ik j k j j=1

j=1, j̸=k

(aik =a⋆ i)

=

n 

n n        aij xLP + ∆xj a ∆x = aij xLP + ij j j j

=

j=1

j=1

j=1 n 

(3)

aij xLP = bi j

j=1

is true for all i ∈ I. But this means that we have found a feasible solution with an objective value of  m  z = j=1 xLP + 1 implying that j   n n  (z ⋆ (E)≥z)   ⋆ ⋆ LP LP zc (E) − z (E) ≤ xj − xj +1 j=1

=

n  

j=1

  −1 < m − 1. xLP − xLP j j

j=1



 Thus, the assertion has been proved for Case 2.2.






This theorem provides first insights on instances having the IRDP and MIRDP, respectively. In particular, Theorem 4 from [2] is contained as a special case. Corollary 2. An instance E = (m, l, L, b) of the skiving stock problem with (i) m = 2 has the IRDP. (ii) m = 3 has the MIRDP.

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Remark 3. Note that instances with m = 3 types of item lengths, in general, do not possess the IRDP. Considering E = (m, l, L, b) = (3, (21, 14, 6), 42, (1, 2, 6)), we notice that ∆(E) = zc⋆ (E) − z ⋆ (E) =

85 43 −1= > 1. 42 42

Thus, E is a non-IRDP instance. Certainly, the upper bound of m − 1 for the gap of the skiving stock somehow appears to be dissatisfying when comparing it with the conjectured constant upper bound of ∆ < 2. But, the authors want the reader to have in mind that, even after several decades of intensive research in the field of the related cutting stock problem (CSP), the best known upper bound for general instances of the CSP is also of O(m), see for m instance [14] or [15, p. 62]. Other results, such as the logarithmic upper bound O (log ( i=1 bi )) [16] have only been proved for some classes with restricted input-data yet; or make use of measures differing from m, like O(log zc⋆ ) in [17]. In particular, it is still an open question whether the so-called MIRUP-conjecture is true for the CSP, cf. to [18] or [19]. Hence, the presented generalization of Zak’s Theorem represents a good starting point for further approaches. 3. The MIRDP of the divisible case In this section, we consider a special class of instances of the skiving stock problem. If an instance E = (m, l, L, b) satisfies li |L for all i ∈ I, then E is an instance of the divisible case. In the context of one-dimensional cutting, those instances have turned out to possess the modified integer round-up property (MIRUP), see [20] or [15, page 24ff.]. In the following, we prove that such a result can be obtained for the skiving stock problem, too. Thereby, the a priori knowledge of the continuous relaxation’s optimal objective value has been proved beneficial. Lemma 4. Let E = (m, l, L, b) be an instance of the divisible case. Then we have zc⋆ (E) =

l⊤ b . L

(4)

Proof. Let ki := L/li be defined for all i ∈ I and let ei ∈ Zm + denote the ith unit vector. Considering the patterns ai = ki · ei and applying them xi = bi /ki times (i ∈ I), we obtain a continuous feasible solution since m  bi aij xj = ki · xi = ki · = bi k i j=1 holds for all i ∈ I. Moreover, the objective value results to m 

m m   bi bi l i l⊤ b z= xj = = = . k L L j=1 j=1 i i=1

Thus, we have found a continuous feasible solution whose objective value equals the upper bound l⊤ b/L proving the optimality of the given feasible solution.  Note that the patterns ai (i ∈ I) from the proof satisfy l⊤ ai = L. Those patterns are called exact patterns. For instances E = (m, l, L, b) of the divisible case, we firstly apply a scaling of the involved lengths such that L is set to L′ = 1, and the (modified) item lengths belong to the set {1/2, 1/3, 1/4, . . .}. Then, the resulting instance is called standard form of E.

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 = (m, κ, 1, b), Lemma 5. Let E = (m, l, L, b) be an instance of the divisible case. Then E is equivalent to E ⊤ where κ = (1/k1 , . . . , 1/km ) is given by ki := L/li for all i ∈ I.  possess the same patterns: Proof. It suffices to show that E and E ⊤ m  a is a pattern of E ⇐⇒ l⊤ a ≥ L, a ∈ Zm + ⇐⇒ κ a ≥ 1, a ∈ Z+ ⇐⇒ a is a pattern of E.

Hence, both corresponding optimization problems are equivalent.



Let E = (m, l, L, b) be an instance of the divisible case in standard form, i.e., L = 1. At first, we only consider the case where E does not contain any exact pattern a ∈ PE⋆ with a ≤ b. Thus, in particular, l i · bi < 1

for all i ∈ I

(5)

holds. Now, the first fit decreasing (FFD) algorithm, proposed in [3] for the dual bin packing problem, represents an initial point of our further investigations. Thereby, bins of capacity lB = L = 1 are filled successively. A bin B is called filled if the objects allocated to it possess a total length of at least lB . If this total length equals lB , the bin B is said to be filled exactly. Furthermore, we say that a bin B can accept an object if, after adding this item, the total length of all objects in B does not exceed lB . Algorithm 1 First Fit Decreasing Phase I: i) Let n ˜ = e⊤ b, with e = (1, . . . , 1)⊤ ∈ Zm + , be the total number of items. Sort them according to decreasing lengths and number them consecutively: 1 > l1⋆ ≥ l2⋆ ≥ . . . ≥ ln⋆˜ . ii) If there are unallocated objects: Let i ∈ {1, . . . , n ˜ } be the index of the first unallocated object. Choose the first bin that is able to accept this object and place it therein. If such a bin does not exist, add a new (empty) bin and place the object therein. Phase II: i) If there is more than one nonempty bin that is not filled: consider the last of these bins, choose one of its items and allocate it to the first of these bins. ii) If there is exactly one nonempty bin that is not filled: allocate its objects to the last bin that is filled.

Note that after Phase II each non-empty bin contains objects with a total length of at least L = 1. Hence, each bin can be interpreted as a uniquely determined packing pattern of E. Actually, observe that all bins but possibly the very last one correspond to a minimal pattern. However, by successively removing appropriate items, also the last non-empty bin can easily be reduced to a minimal pattern. Lemma 6. Let E = (m, l, 1, b) be an instance of the divisible case in standard form not possessing any exact pattern a ∈ PE⋆ with a ≤ b. Furthermore, let q ∈ N denote the number of existing bins after Phase I of Algorithm 1. Then, for all j ∈ {1, . . . , q − 1}, the jth bin Bj contains at least j + 1 objects after the completion of Phase I. Proof. Let j ∈ {1, . . . , q − 1} be given and let lij ∈ {l1 , . . . , lm } denote an object length that is contained in bin Bj . Then, we claim that lij ≤

1 j+1

(6)

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holds: Due to (5), Phase I of Algorithm 1 needs at most j − 1 bins for the allocation of all objects with li⋆ ≥ 1/j. Hence, all objects with lengths li ≥ 1/j are contained in the bins B1 , . . . , Bj−1 implying that the jth bin Bj only possesses objects with lengths ≤ 1/(j + 1). Since we have j < q, there exists a bin Bj+1 . Based on the same considerations, all of its objects’ lengths are ≤ 1/(j + 2). Let l⋆ be the largest of these objects lengths, i.e., l⋆ ≤ 1/(j + 2) holds particularly. Due to Phase I of Algorithm 1, the remaining capacity R(j) of Bj results to R(j) < l⋆ ≤

1 j+2

(7)

since, otherwise, the object with length l⋆ would have been assigned to Bj . j·

Assuming that there are at most j objects in Bj , then, due to (6), their total length would be at most 1 j+1 , i.e., the remaining capacity of Bj would result to R(j) ≥ 1 − j ·

1 1 1 = > j+1 j+1 j+2

which gives the contradiction, see (7). Thus, our assumption was wrong and the lemma has been proved.



The following lemma states a lower bound for the number q of bins that exist after Phase I of Algorithm 1. Thereby, the assumption that E does not contain any exact pattern a ∈ PE⋆ with a ≤ b plays an important role.   Lemma 7. Let E = (m, l, 1, b) be an instance of the divisible case in standard form with K := l⊤ b for some K ∈ N. If E does not contain any exact pattern a ∈ PE⋆ with a ≤ b the inequality q ≥ K + 1 holds. Proof. We assume that there were at most K bins after the completion of Phase I. Then, the total length of all objects would be strictly smaller than K since no bin is filled exactly. This gives the contradiction.  Now we have all necessary preliminaries to prove our main result. Theorem 8. Let E be an instance of the divisible case in standard form not containing any exact pattern a ∈ PE⋆ with a ≤ b. Then E has the MIRDP.   Proof. Let q denote the number of existing bins after Phase I. Then q ≥ l⊤ b + 1 holds. Let aij be the number of items of type i ∈ I that have been assigned to the bin j ∈ {1, . . . , q}. Due to the assumption of m this theorem, i=1 li aij < 1 holds for all j ∈ {1, . . . , q}. Let liq ∈ {l1 , . . . , lm } be the length of an arbitrary object of bin Bq . Then, we obtain ∀ j ∈ {1, . . . , q − 1} : 1 −

m 

li aij < liq

(8)

i=1

since, otherwise, the considered object would have been assigned to a previous bin. Case 1: Bq contains at least q − 1 objects. Due to (8), each one of these q − 1 objects can be assigned to one of the bins B1 , . . . , Bq−1 leading to a   total length of at least L = lB in each of these bins. Since q ≥ l⊤ b + 1 holds, we found a feasible solution ⊤    for the ILP with objective value z = q − 1 ≥ l b . In this case, we can even state z ⋆ (E) = l⊤ b leading to the IRDP of E which implies the MIRDP, too.

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Case 2: Bq contains k < q − 1 objects. By means of the same procedure as in the first case, the bins B1 , . . . , Bk can be filled. Now, we consider the bin Bq−1 . Let liq−1 be an arbitrary item length that is contained in Bq−1 . Then, in analogy to (8), we obtain 1−

m 

li aij < liq−1

i=1

for all j = k + 1, . . . , q − 2 since, otherwise, this object would have been assigned to a previous bin by Phase I of the algorithm. Due to Lemma 6, Bq−1 contains at least q objects. As described in the first case,   they can be used to fill the bins Bk+1 , . . . , Bq−2 . Since q ≥ l⊤ b + 1 holds, we found a feasible solution     for the ILP with objective value z = q − 2 ≥ l⊤ b − 1. Hence, we obtain z ⋆ (E) ≥ l⊤ b − 1 proving the MIRDP.  The previous investigations could only cope with special instances of the divisible case since we required the absence of exact patterns with a ≤ b for our argumentations. In the following, we show how Theorem 8 can be applied to arbitrary instances of the divisible case. Theorem 9. Let E = (m, l, 1, b) be an instance of the divisible case in standard form. Then E has the MIRDP. Proof. We divide E into two subinstances E1 = (m, l, 1, α) and E2 = (m, l, 1, β). Therefore, we remove successively exact patterns a ∈ PE⋆ with a ≤ b from E and assign their corresponding items to E2 . The remaining instance, without exact patterns satisfying a ≤ b, is called E1 . Note that this partition is not unique. For the sake of simplicity, we allow αi and βi to equal zero for some i ∈ I thus obtaining m item types in both subinstances. In E1 we can apply Theorem 8 since E1 matches all of its prerequisites, and     obtain z1 ≥ l⊤ α − 1. Moreover, E2 consists of (disjoint) exact patterns, which leads to l⊤ β = l⊤ β and z2 = l⊤ β. Since E1 and E2 share no common objects, a feasible solution of E can be obtained by combining the feasible solutions of the subinstances. This results to an objective value of    (l⊤ β ∈ Z+ )  ⊤ (α+β=b)  ⊤  = l α + l⊤ β − 1 = l b − 1, l⊤ α − 1 + l⊤ β   i.e., the optimal objective values satisfy z ⋆ (E) ≥ l⊤ b − 1 proving the MIRDP.  z = z1 + z2 ≥



Obviously, this result cannot be improved since there are instances of the divisible case not possessing the IRDP, cf. Remark 3 at the end of Section 2. 4. A construction principle for instances with gaps greater than one In the previous explanations, we proved the MIRDP for the divisible case, but only indicated a single example of a non-IRDP instance. Thus, the question arises whether there are many of those instances at all. As an additional motivation for this investigation, let us recapitulate briefly the chronology of the gap for the related cutting stock problem. For a period of more than 20 years, it was conjectured that the gap of the CSP can be bounded above by the constant 1. The first counterexample to this claim (a so-called non-IRUP instance) was found by Marcotte [13], but the corresponding instance made use of input-data in the range of 106 . Hence, the theory of the gap of the CSP was still considered to be rather unimportant, particularly due to its practical background where such unusual data may not occur. Some years later, Fieldhouse [21] constructed the first counterexample of moderately sized input-data, but this phenomenon was still considered as exceptional due to the rarity of non-IRUP instances. Finally, an infinite number of such instances could be presented, see for instance [22], and the research on the theory of the gap of the CSP was intensified.

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In this section, we present a construction principle for an infinite number of non-equivalent non-IRDP instances and calculate their gaps explicitly. As an initial point, we state the following necessary condition. Lemma 10. Let E = (m, l, 1, b) be an instance of the divisible case in standard form. If ∆(E) ≥ 1 holds, then   K := K(E) := l⊤ b ≥ 2 is true. Proof. For K ∈ {0, 1} we have z ⋆ (E) = K and ∆(E) = zc⋆ (E) − z ⋆ (E) < (K + 1) − K = 1.



Consequently, we can restrict our investigations to instances with K ≥ 2. So, let K ∈ N with K ≥ 2 be given and let P denote the set of all prime numbers, then we define     q       p−1  (9) ∈ [K − 1, K) p¯ := p¯(K) := inf q ∈ P     p=2, p    p∈P and Λ := Λ(K) := {p ∈ P | p ≤ p¯ }. Obviously, p¯ is well-defined due to the divergence of the defining series. Since all denominators of the considered summands {1/p | p ∈ Λ } are relatively prime, the common denominator  h := h(K) := p (10) p∈Λ

is given by the product of all these primes. Hence, the construction of p¯ immediately implies that there exists s := s(K) ∈ Z+ with s < h and  p−1 s =K −1+ . (11) p h p∈Λ

The proofs of the following results (Lemma 11, Proposition 12, Lemma 13) are very technical; hence they can be found online in our preprint [23]. Lemma 11. Let h and s be given as in (10) and (11). Then, the following statements are true: (i) s and h are coprime. (ii) s > 0. (iii) h − s > 1. In what follows, we would like to construct an instance E = (m, l, 1, b) of the divisible case in standard form satisfying ⌊l⊤ b⌋ = K. Due to (11), our current item supply is not yet sufficient. Thus, we have to add some items in a smart way. Proposition 12. There are positive integers v, w ∈ N satisfying K −1+

s v 1 + =K+ . h w hw

(12)

In the following lemma, we state some properties of v and w that will be needed for the construction of non-IRDP instances. Lemma 13. The following statements are true: (i) Let C ∈ N be a given constant. Then, w in (12) can be chosen such that w > C holds. (ii) The numbers w in (12) and h in (10) are coprime.

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(iii) The numbers v and w in (12) can be chosen such that v < w holds. (iv) There exists a sequence {(v(K), w(K))}K≥2 in N2 with: (a) The element (v(K), w(K)) satisfies (12) for all K ≥ 2. (b) The sequence {w(K)}K≥2 is monotonically increasing with w(2) > 1. (c) v(K) < w(K) holds for all K ≥ 2. Definition 1. A sequence possessing the properties of Lemma 13(iv) is called M -sequence. With the help of these preliminary studies, we can formulate and prove the main result of this section.    w(K)  Theorem 14. Let K ≥ 2 and an M -sequence v(K), be given. Setting  K≥2   1 1 1 1 1 L = 1, m = |Λ(K)| + 1, l= , , ,..., , , b = (1, 2, 4, . . . , p¯(K) − 1, v(K)) , 2 3 5 p¯(K) w(K) we obtain an instance E = (m, l, L, b) possessing the following properties:   (i) E is an instance of the divisible case in standard form with l⊤ b = K. (ii) There is no pattern a ∈ Zm + with a ≤ b and 1 ≤ l⊤ a < 1 +

1 . h(K)w(K)

(13)

(iii) The gap can be calculated by ∆(E) = 1 +

1 , h(K)w(K)

(14)

i.e., E is a non-IRDP instance. Proof.

(i) Due to L = 1, E represents an instance of the divisible case in standard form. Besides, we have  p−1 v(K) (11) s(K) v(K) (12) 1 + = K −1+ + = K+ l⊤ b = p w(K) h(K) w(K) h(K)w(K) p∈Λ(K)

implying that ⌊l⊤ b⌋ = K. (ii) Let a ∈ Zm + be a pattern satisfying (13). Obviously, the common denominator of all components of the vector l is given by h(K)w(K). Hence, a pattern can only have a total length of 1+

ξ h(K)w(K)

for some ξ ∈ Z+ . According to the presumption (13), only the case ξ = 0 is possible. Thus, it suffices to show that an exact pattern a ∈ PE⋆ with a ≤ b does not exist. ⊤ Assuming that a ∈ Zm + is an exact pattern with a ≤ b, i.e., l a = 1 holds. Let I(a) = {i ∈ I | ai > 0 } denote the set of active indices. Then, l⊤ a = 1 ⇐⇒

m 

ai li = 1 ⇐⇒

i=1

1  ai ri (a) = 1 h(a) i∈I(a)



holds where h(a) := j∈I(a) 1/lj denotes the common denominator of all lengths appearing in a ∈ Zm + and ri (a) is given by  1 ri (a) := = li h(a) ∈ N lj j∈I(a), i̸=j

for all i ∈ I(a).

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 In order to obtain l⊤ a = 1, the numerator i∈I(a) ai ri (a) has to be a multiple of every denominator in the set N (a) := {1/li | i ∈ I(a) }. Let t ∈ N (a) be given, then there is j ∈ I(a) with t = lj−1 . Consequently, t divides all summands of the numerator with i ̸= j since, in these cases, t divides ri (a). However, t is not a divisor of aj rj (a) since, on the one hand, aj ≤ bj < lj−1 = t holds, and, on the other hand, rj (a) and lj−1 are coprime. Thus, the numerator is not a multiple of t implying that there does not exist any exact pattern a ∈ Zm + with a ≤ b. (iii) Due to Lemma 4 and part (i) of this theorem, the considered instance E has the optimal objective value 1 . zc⋆ (E) = l⊤ b = K + h(K)w(K) Due to Theorem 9, we also have ∆(E) < 2 entailing z ⋆ (E) ∈ {K − 1, K}. Hence, it suffices to prove that z ⋆ (E) ̸= K is true. For the sake of contradiction, let us assume that there are minimal K patterns a1 , . . . , aK ∈ PE⋆ such that the resource constraint j=1 aj ≤ b is respected. Note that, obviously, aj ≤ b (j = 1, . . . , K) has to hold, and, therefore, statement (ii) of this theorem can be applied. Due to K ≥ 2, this leads to   K  (ii) K 1 1 ⊤ ⊤ j =K+ >K+ = l⊤ b l b≥ l a ≥ K · 1+ h(K)w(K) h(K)w(K) h(K)w(K) j=1 which gives the contradiction. Hence, we have z ⋆ (E) = K − 1, and the gap ∆(E) results to   1 1 ∆(E) = zc⋆ (E) − z ⋆ (E) = K + − (K − 1) = 1 + h(K)w(K) h(K)w(K) proving (14). In particular, E is a non-IRDP instance.



We would like to point out that this construction can be used in a more general way to obtain a multitude of further non-IRDP instances.  ⊆ P a subset of the prime numbers and Remark 15. Let K ∈ N be a natural number with K ≥ 2, P   f : P → N a mapping with f (p) ≤ p − 1 for all p ∈ P such that the series  f (p) p p∈ P diverges. Defining, in analogy to (9),     f, P f, P  p¯ := p¯ (K) := inf q ∈ P   

      q   f (p)  ∈ [K − 1, K)    p=2, p    p∈P

the whole theory presented in this section can be applied in a straightforward way to obtain non-IRDP instances.  = P as well as f (p) = p − 1 for all p ∈ P.  Hence, we took the greatest possible Note that we used P denominators and also their maximum availability. Thus, for some fixed K ≥ 2, the instances from Theorem 14 contain the minimum number m of item types among all instances that can be generated by this method. Based on the results of Theorem 14 it is possible to obtain instances of the divisible case with gaps arbitrarily close to 22/21. Remarkably, the best known gap 137/132 (see [19, p. 69]) for the divisible case of the related

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and extensively studied cutting stock problem can be surpassed thereby. As an initial point, note that due to monotonicity arguments the largest gap of Theorem 14 is provided by the case K = 2, i.e., by the instance     1 1 1 ⋆ E = 3, , , , 1, (1, 2, 6) 2 3 7 which is known from Remark 3, and results to ∆(E ⋆ ) = 1 + 1/42. By adding an item type this gap can be increased as to be noticed in the following exemplary series of instances: Corollary 16. The instance E ⋆ (d) :=

    1 1 1 1 , , , d+1 , 1, (1, 2, 6, 2d+1 − 1) 4, 2 3 7 2 · 42

with d ∈ N provides the gap ∆(E ⋆ (d)) =

1 1 22 − · , 21 2d+1 42

i.e., we have limd→∞ ∆(E ⋆ (d)) = 22/21. Proof. Let d ∈ N be fixed. Obviously, the given instance E ⋆ (d) belongs to the divisible case leading to 1 2 6 2d+1 − 1 1 43 1 1 + + + · = − · . 2 3 7 2d+1 42 21 2d+1 42 Due to the definition of the gap, only z ⋆ (E ⋆ (d)) = 1 remains to show. Since z ⋆ (E ⋆ (d)) ∈ {1, 2} clearly holds, it suffices to prove that there is no minimal pattern a ∈ P ⋆ (E ⋆ (d)) with 1 ≤ l⊤ a < 22/21: For the sake of contradiction, let a⋆ denote such a pattern. Due to    1 z¯ := min l⊤ a  a ∈ Z4+ , a4 = 0, l⊤ a ≥ 1, a ≤ b = 1 + , 21 zc⋆ (E(d)) =

we conclude a⋆4 ≥ 1. Consequently, we have l⊤ (a⋆1 , a⋆2 , a⋆3 , 0)⊤ < 1 by minimality of a⋆ , and also l⊤ (a⋆1 , a⋆2 , a⋆3 , 0)⊤ = l⊤ a⋆ − l4 a⋆4 ≥ 1 − l4 a⋆4 ≥ 1 −

1 2d+1 − 1 1 · >1− . d+1 2 42 42

But the latter contradicts to    1 z := max l⊤ a  a ∈ Z4+ , a4 = 0, l⊤ a ≤ 1, a ≤ b = 1 − , 42 which proves the assertion.  Note that this approach of adding appropriate item types in order to increase the gap of a given instance can be generalized. For more details the interested reader is referred to our preprint [23, Theorem 16]. 5. Conclusion and outlook In this paper, we gave an introduction to the gap of the skiving stock problem from a theoretical point of view. As a starting point, we generalized a result of [2] in order to obtain a general upper bound for the gap depending on the number of item types. Thereby, we also stated a first sufficient condition for instances possessing the MIRDP. Thereupon, we studied the divisible case intensively. For these instances, we have proved, as a main result, the MIRDP by means of the FFD heuristic for the dual bin packing problem. Moreover, we presented a construction principle obtaining an infinite number of non-equivalent non-IRDP instances of the divisible case, and showed how this method can be used to obtain instances with gaps arbitrarily close to 22/21 which surpasses the best known gap for the divisible case of the related cutting stock problem.

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Except for the continuous relaxation studied in this paper, there are other possible relaxations that are of theoretical and practical interest. One of these is the proper relaxation where, as an additional restriction, only minimal patterns a ∈ PE⋆ with a ≤ b, so-called proper patterns, are considered. This relaxation has also been studied in one-dimensional cutting scenarios [24,25], so its practicability in terms of the skiving stock problem will be investigated in the future. Since we mainly focused on the divisible case it will be part of our future research to study the gap of arbitrary instances of the skiving stock problem. Therefore, we aim at improving the upper bound m − 1 introduced in this paper in order to get a bit closer to the MIRDP-conjecture. Acknowledgments The authors would like to thank two anonymous referees for very valuable remarks on an earlier draft. References [1] M. Johnson, C. Rennick, E. Zak, Skiving addition to the cutting stock problem in the paper industry, SIAM Rev. 39 (3) (1997) 472–483. [2] E. Zak, The skiving stock problem as a counterpart of the cutting stock problem, Int. Trans. Oper. Res. 10 (2003) 637–650. [3] S. Assmann, D. Johnson, D. Kleitman, J.-T. Leung, On a dual version of the one-dimensional bin packing problem, J. Algorithms 5 (1984) 502–525. [4] M. Labbe, G. Laporte, S. Martello, An exact algorithm for the dual bin packing problem, Oper. Res. Lett. 17 (1995) 9–18. [5] M. Peeters, Z. Degraeve, Branch-and-price algorithms for the dual bin packing and maximum cardinality bin packing problem, European J. Oper. Res. 170 (2) (2006) 416–439. [6] J. Csirik, J. Frenk, G. Galambos, A. Rinnooy Kan, Probabilistic analysis of algorithms for dual bin packing problems, J. Algorithms 12 (1991) 189–203. [7] J. Bruno, P. Downey, Probabilistic bounds for dual bin-packing, Acta Inform. 22 (1985) 333–345. [8] A. Alvim, C. Ribeiro, F. Glover, D. Aloise, A hybrid improvement heuristic for the one-dimensional bin packing problem, J. Heuristics 10 (2) (2004) 205–229. [9] B. Vijayakumar, P. Parikh, R. Scott, A. Barnes, J. Gallimore, A dual bin-packing approach to scheduling surgical cases at a publicly-funded hospital, European J. Oper. Res. 224 (3) (2013) 583–591. [10] P. Gilmore, R. Gomory, A linear programming approach to the cutting-stock problem (part i), Oper. Res. 9 (1961) 849–859. [11] J. Martinovic, G. Scheithauer, Integer linear programming models for the skiving stock problem, European J. Oper. Res. 251 (2) (2016) 356–368. [12] S. Baum, L. Trotter, Integer rounding for polymatroid and branching optimization problems, SIAM J. Algebr. Discrete Methods 2 (1981) 416–425. [13] O. Marcotte, Topics in combinatorial packing and covering, (Technical Report No.568), Cornell University, 1983. [14] C. Filippi, On the bin packing problem with a fixed number of object weights. Technical report, University of Padova, 2007. [15] J. Rietz, Untersuchungen zu MIRUP f¨ ur Vektorpackprobleme (Ph.D. thesis), TU Bergakademie Freiberg, 2003. [16] F. Eisenbrand, D. P´ alv¨ olgyi, T. Rothvoß, Bin packing via discrepancy of permutations, ACM Trans. Algorithms 9 (3) (2013) Article 24. [17] R. Hoberg, T. Rothvoss, A logarithmic additive integrality gap for bin packing, 2015. (arxiv:1503.08796v1). [18] G. Nemhauser, Column generation for linear and integer programming, in: Gr¨ otschel, M., (Ed.), Optimization Stories: 21st International Symposium on Mathematical Programming. Documenta Mathematica, 2012. [19] G. Scheithauer, Zuschnitt- und Packungsoptimierung – Problemstellungen, Modellierungstechniken, L¨ osungsmethoden, first ed., Vieweg+Teubner, Wiesbaden, 2008. [20] G. Scheithauer, J. Terno, About the gap between the optimal value of the integer and continuous relaxation one-dimensional cutting stock problem, in: Operations Research Proceedings, Springer Verlag, Berlin und Heidelberg, 1992, pp. 439–444. [21] M. Fieldhouse, The duality gap in trim problems, SICUP-Bull. 5 (1990) 4–5. [22] J. Rietz, G. Scheithauer, Families of non-irup instances of the one-dimensional cutting stock problem, Discrete Appl. Math. 121 (2002) 229–245. [23] J. Martinovic, G. Scheithauer, Integer rounding and modified integer rounding for the skiving stock problem. Preprint MATH-NM-02-2015, Technische Universit¨ at Dresden, 2015. [24] C. Nitsche, G. Scheithauer, J. Terno, Tighter relaxations for the cutting stock problem, European J. Oper. Res. 112 (3) (1999) 654–663. [25] V. Kartak, A. Ripatti, G. Scheithauer, Minimal proper non-IRUP instances of the one-dimensional cutting stock problem, Discrete Appl. Math. 187 (2015) 120–129.