Integer Programming and m-irreducibility of numerical semigroups
aa r X i v : . [ m a t h . O C ] J a n INTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICALSEMIGROUPS V´ICTOR BLANCO AND JUSTO PUERTO
Abstract.
This paper addresses the problem of decomposing a numerical semigroup into m -irreduciblenumerical semigroups. The problem originally stated in algebraic terms is translated, introducingthe so called Kunz-coordinates, to resolve a series of several discrete optimization problems. First,we prove that finding a minimal m -irreducible decomposition is equivalent to solve a multiobjectivelinear integer problem. Then, we restate that problem as the problem of finding all the optimalsolutions of a finite number of single objective integer linear problems plus a set covering problem.Finally, we prove that there is a suitable transformation that reduces the original problem to find anoptimal solution of a compact integer linear problem. This result ensures a polynomial time algorithmfor each given multiplicity m . We have implemented the different algorithms and have performedsome computational experiments to show the efficiency of our methodology. Introduction
The rich literature of discrete mathematics contains an important number of references, of theo-retical results, that have helped in solving or advancing in the resolution of many different discreteoptimization problems. Nowadays, it is considered a standard to cite the connections between graphtheory, commutative algebra and optimization. For instance, it is a topic to mention the connectionsbetween the results by K¨onig and Egervary in graph theory and the algorithms by Kuhn for the assign-ment problem or by Edmonds for the maximum matching problem ([14, 15, 20, 19, 24]). In the samevein, there are well-known papers that apply algebraic tools to solve single objective integer linearproblems (see [9], [11], [22]) or multiobjective integer linear problems (see [3, 5, 4, 12]). Moreover,more recently we can find a rich body of literature (see [25] and the references therein) that addressesgeneral discrete optimization problems (non necessarily linear) using tools borrowed from pure andapplied Algebra (Graver bases and the like).On the other hand, although less known, there have been also some applications of integer program-ming to solve problems of commutative algebra (see [1, 2, 7]). The goal of this paper is to analyze andsolve another problem arising in commutative algebra using tools from integer programming.A numerical semigroup is a subset S of Z + (here Z + denotes the set of non-negative integers)closed under addition, containing zero and such that Z + \ S is finite. Numerical semigroups werefirst considered while studying the set of nonnegative solutions of Diophantine equations and theirstudy is closely related to the analysis of monomial curves (see [13]). By these reasons, the theory ofnumerical semigroups has attracted a number of researchers from the algebraic community. This facthas motivated that some of the terminology used in Algebraic Geometry has been exported to thisfield. For instance, the multiplicity, the genus, or the embedding dimension of a numerical semigroup.Further details about the theory of numerical semigroups can be found in the recent monograph byRosales and Garc´ıa-S´anchez [32].In recent years, the problem of decomposing numerical semigroups into irreducible ones has attractedthe interest of the research community (see [8, 27, 29, 30, 31]). Recall that a numerical semigroupis irreducible if it cannot be expressed as an intersection of two numerical semigroups containing itproperly. Furthermore, more recently a different notion of irreducibility, the m -irreducibility ([6]) has Date : August 20, 2018.2010
Mathematics Subject Classification.
Key words and phrases. integer programming, numerical semigroups, irreducibility, multiplicity. appeared and has started to be analyzed. A numerical semigroup with multiplicity m is said m -irreducible if it cannot be expressed as an intersection of two numerical semigroups with multiplicity m and containing it properly. (Recall that the multiplicity of a numerical semigroup is the smallestnon zero element belonging to it.) The question of existence of m -irreducible decompositions has beenproved in [6]. Nevertheless, it is still missing a methodology, different to the almost pure brute forceenumeration, to find m -irreducible decompositions of minimal size.In this paper, we give a methodology to obtain such a minimal decomposition into m -irreduciblenumerical semigroups by using tools borrowed from discrete optimization. To this end, we identifyone-to-one numerical semigroups with the integer vectors inside a rational polytope (see [26]). Forthe sake of this identification, we introduce the notion of Kunz-coordinates vector to translate theconsidered problem in the problem of finding some integer optimal solutions, with respect to appro-priate objective functions, in the Kunz polytope. Then, the problem of enumerating the minimal m -irreducible numerical semigroups involved in the decomposition is formulated as a multiobjectiveinteger program. We state that solving this problem is equivalent to enumerate the entire sets ofoptimal solutions of a finite set of single-objective integer problems. The number of integer problemsto be solved is bounded above by m −
1, where m is the multiplicity of the semigroup to be decom-posed. Finally, we solve a set covering problem to ensure that the decomposition has the smallestnumber of elements. Although this approach is exact its complexity is rather high and in general onecannot prove that it is polynomial for any given multiplicity m . This comes from the fact that thereare nowadays relatively few exact methods to solve general multiobjective integer and linear problems(see [16]) and it is known that the complexity of solving in general this type of problems is m -irreducible decomposition is polynomially solvable.In Section 2 we recall the main definitions and results needed for this paper to be selfcontained.Section 3 is devoted to translate the problem of finding numerical semigroups of a given multiplicityinto the problem of detecting integer points inside a rational polytope, introducing the notion of Kunz-coordinates vector. We write in Section 4 the conditions, in terms of the Kunz-coordinates vector foa numerical semigroup to be a m -irreducible oversemigroup. Section 5 is devoted to formulate theproblem of decomposing and minimally decomposing into m -irreducible numerical semigroups as amathematical programming problem. We give an exact and a heuristic approach for computing such aminimal decomposition based on solving some integer programming problems. In Section 6 we presenta compact model to compute, by solving only one integer programming problem, a minimal decomposi-tion of a numerical semigroup into m -irreducible numerical semigroups. There, we also prove that thisproblem is polynomially solvable. Finally, in Section 7 we show some computational tests performedto check the efficiency of the presented algorithms with respect to the current implementation in GAP [10]. 2.
Preliminaries
For the sake of readability, in this section we recall the main results about numerical semigroupsneeded so that the paper is selfcontained.Let S be a numerical semigroup. We say that { n , . . . , n p } is a system of generators of S if S = { p X i =1 n i x i : x i ∈ Z + , i = 1 , . . . , p } . We denote S = h n , . . . , n p i if { n , . . . , n p } is a system of generatorsof S .The least positive integer belonging to S is denoted by m( S ), and is called the multiplicity of S (m( S ) = min( S \ { } )).Two important notions of irreducibility are extensively used through this paper. They are thefollowing: Definition 1 (Irreducibility and m-irreducibility) . NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 3 • A numerical semigroup is irreducible if it cannot be expressed as an intersection of two nu-merical semigroups containing it properly. • A numerical semigroup with multiplicity m is m - irreducible if it cannot be expressed as anintersection of two numerical semigroups with multiplicity m containing it properly. In [6] the authors analyze and characterize the set of m -irreducible numerical semigroups. Notethat, in particular, any irreducible numerical semigroup is m -irreducible, while the converse is nottrue. One of the results in that paper is the key for the analysis done through this paper and it isstated as follows. Proposition 2 ([6]) . Let S be a numerical semigroup with multiplicity m . Then, there exist S , . . . , S k m -irreducible numerical semigroups such that S = S ∩ · · · ∩ S k . From the above result one may think of obtaining the minimal number of elements involved in theabove intersection of m -irreducible numerical semigroup. Formally, we describe what we understand bydecomposing and minimally decomposing a numerical semigroup with multiplicity m into m -irreduciblenumerical semigroups. Definition 3 (Decomposition into m -irreducible numerical semigroups) . Let S be a numerical semi-group with multiplicity m . Decomposing S into m -irreducible numerical semigroups consists of findinga set of m -irreducible numerical semigroups S , . . . , S r( S ) such that S = S ∩ · · · ∩ S r( S ) (This decom-position is always possible by Proposition 2).A minimal decomposition of S into m -irreducible numerical semigroups is a decomposition withminimum r (S) (minimal cardinality of the number of m -irreducible numerical semigroups involved inthe decomposition). For a numerical semigroup S , the set of gaps of S , G( S ), is the set Z + \ S (that is finite by definitionof numerical semigroup). We denote by g( S ) the cardinal of that set, that is usually called the genus of S . The Frobenius number of S , F( S ), is the largest integer not belonging to S .Let S be a numerical semigroup with multiplicity m . To decompose S into m -irreducible numericalsemigroups, we first need to know how to identify those m -irreducible numerical semigroups. In [6]it is proved that S is m -irreducible if and only if it is maximal (w.r.t. the inclusion order) in the setof numerical semigroups with multiplicity m and Frobenius number F( S ). In [29] it is stated that anumerical semigroup, S , is irreducible if and only if g( S ) = (cid:24) F( S ) + 12 (cid:25) . Next, we recall two resultsin [6] that characterize, in terms of the genus and the Frobenius number, the set of m -irreduciblenumerical semigroups. Proposition 4 ([6]) . A numerical semigroup with multiplicity m , S , is m -irreducible if and only ifone of the following conditions holds: (1) If F( S ) = g( S ) = m − then S = { x ∈ Z + : x ≥ m } ∪ { } . (2) If F( S ) ∈ { m + 1 , . . . , m − } and g( S ) = m then S = { x ∈ Z + : x ≥ m, x = F( S ) } ∪ { } . (3) If F( S ) > m then S is an irreducible numerical semigroup. (In this case g( S ) = (cid:24) F( S ) + 12 (cid:25) ). Corollary 5 ([6]) . Let S be a numerical semigroup with multiplicity m . Then, S is m -irreducible ifand only if g( S ) ∈ (cid:8) m − , m, (cid:24) F( S ) + 12 (cid:25) (cid:9) . For a given numerical semigroup, S , our goal is to find a set of m -irreducible numerical semigroupswhose intersection is S . Then, we can restrict the search of these semigroups to the set of numericalsemigroups that contain S . This set is called the set of oversemigroups of S . Definition 6 (Oversemigroups) . Let S be a numerical semigroup with multiplicity m . The set, O ( S ) ,of oversemigroups of S is O ( S ) = { S ′ numerical semigroup : S ⊆ S ′ } The set, O m ( S ) , of oversemigroups of S with multiplicity m is O m ( S ) = { S ′ ∈ O ( S ) : m( S ′ ) = m } . V. BLANCO AND J. PUERTO
Denote by I m ( S ) the set of minimal m -irreducible numerical semigroups, with respect to the in-clusion poset, in the set O m ( S ). From the set I m ( S ) we can obtain, a first decomposition of S into m -irreducible numerical semigroup, although, in general, it is not minimal (see Example 27 in [6]). Lemma 7.
Let S be a numerical semigroup with multiplicity m and I m ( S ) = { S , . . . , S n } . Then S = S ∩ · · · ∩ S n is a decomposition of S into m -irreducible numerical semigroups. Clearly, this basic decomposition is not ensured to be minimal since it may use redundant elements.
Remark 8.
Note that if ˆ S is a numerical semigroup with multiplicity m , by Proposition 4, g( ˆ S ) = m − if and only if ˆ S = { , m, →} ( → denotes that every integer greater than m belongs to ˆ S ). Hence, this m -irreducible numerical semigroup only appears in its own decomposition and in no one else.This is due to the fact that ˆ S = { , m, →} is the maximal element in the set of numerical semigroupswith multiplicity m , and then O m ( ˆ S ) = I m ( ˆ S ) = { ˆ S } (see [6] for further details). From now on, we assume that S = ˆ S = { , m, →} since by the above remark, the decomposition ofˆ S is trivial.By Proposition 4 and Remark 8, if S = ˆ S = { , m, →} , its decomposition into m -irreduciblenumerical semigroups uses two types of numerical semigroups: those that are irreducible (g( S ) = (cid:24) F( S ) + 12 (cid:25) ) and those that have genus equal the multiplicity of S .To refine the search of the elements in I m ( S ), first, we need to introduce the notion of special gap. Definition 9.
Let S be a numerical semigroup. The special gaps of S are the elements in the followingset: SG( S ) = { h ∈ G( S ) : S ∪ { h } is a numerical semigroup } where G( S ) is the set of gaps of S . We denote by SG m ( S ) = { h ∈ SG( S ) : h > m } . In [6], the authors proved that S is m -irreducibleif and only if m ( S ) ≤ A stands for the cardinality of the set A ). Moreover, SG m ( S ) = ∅ ifand only if S = { , m, →} (there are no gaps greater than m in S ).Also, if we know the special gaps of a numerical semigroup, we can search for its decomposition byusing the following result. Proposition 10 ([6]) . Let
S, S , . . . , S n be numerical semigroups with multiplicity m . S = S ∩· · ·∩ S n if and only if SG m ( S ) ∩ (G( S ) ∪ · · · ∪ G( S n )) = SG m ( S ) . From the above proposition, even if the minimal m -irreducible numerical semigroups are known, I m ( S ) = { S , . . . , S m } , some of these elements may be discarded when looking for a minimal m-irreducible decomposition, by checking if there are redundant elements in the intersection SG m ( S ) ∩ (G( S ) ∪ · · · ∪ G( S n )).Then, the key is to choose elements in I m ( S ) that minimally cover the special gaps of S . To thisfor, we may solve a problem fixing each of the special gaps to be covered. Note that an upper boundof the number of problems to be solved is the number of special gaps of a numerical semigroup that isbounded above by m − Lemma 11.
Let S = { , m, →} be a numerical semigroup with multiplicity m , and h ∈ SG m ( S ) . Then,there exists a minimal decomposition of S into m -irreducible numerical semigroups, S = S ∩ · · · ∩ S n such that, either h = F( S i ) for some i or h S i for some i such that there exists h ′ ∈ SG m ( S i ) with F( S i ) = h ′ > h .Proof. By Proposition 2, there exists a minimal decomposition of S into m -irreducible numericalsemigroup, S = S ∩· · ·∩ S k . By applying Proposition 10, this decomposition must verify that SG m ( S ) ∩ (G( S ) ∪ · · · ∪ G( S n )) = SG m ( S ). Each special gap h ∈ SG m ( S ) must be in G( S i ) for some i =1 , . . . , n . Assume that h = F( S i ) and that for all h ′ ∈ SG m ( S i ) with h ′ > h , F( S i ) = h ′ . Then, S ′ i = S i ∪{ F( S i ) } is a m -irreducible numerical semigroup such that SG m ( S ) ∩ (G( S ) ∪ · · · G( S ′ i ) · · · ∪ G( S n )) = NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 5 SG m ( S ). Then, we have obtained a different minimal decomposition. (Note that it has the samenumber of terms than the original one.)By repeating this procedure for each h ∈ SG m ( S ) while it is possible, we find a minimal decompo-sition of S fulfilling the conditions of the lemma. (cid:3) The Kunz-coordinates vector
The analysis done through this paper uses mathematical programming tools to solve the problem ofdecomposing a numerical semigroup into m -irreducible numerical semigroups. For the sake of trans-lating the problem to a discrete optimization problem, we use an alternative encoding of numericalsemigroups different from the system of generators. We identify each numerical semigroup with multi-plicity m with a nonnegative integer vector with m − m is the multiplicity of thesemigroup. To describe this identification we first need to give the notion of Ap´ery set of a numericalsemigroup with respect to its multiplicity Definition 12.
Let S be a numerical semigroup with multiplicity m . The Ap´ery set of S with respectto m is the set Ap(
S, m ) = { s ∈ S : s − m S } . However we are interested in the following characterization of the Ap´ery set (see [32]): Let S be anumerical semigroup with multiplicity m , then Ap( S, m ) = { w , w , . . . , w m − } , where w i is theleast element in S congruent with i modulo m , for i = 1 , . . . , m − S, m ) completely determines S , since S = h Ap(
S, m ) ∪ { m }i (see [26]), andthen, we can identify S with its Ap´ery set with respect to its multiplicity. Moreover, the set Ap( S, m )contains, in general, more information than an arbitrary system of generators of S . For instance,Selmer in [33] gives the formulas, g( S ) = m (cid:16)P w ∈ Ap(
S,m ) w (cid:17) − m − and F( S ) = max(Ap( S, m )) − m .Moreover, one can test if a nonnegative integer s belongs to S by checking if w s (mod m ) ≤ s . Thenotion of Ap´ery set is also given when we consider any n ∈ S instead of m , rewriting then the definitionadequately (see [32]). Moreover, the smallest Ap´ery set is Ap( S, m ).We consider an slightly but useful modification of the Ap´ery set that we call the
Kunz-coordinatesvector . Definition 13 (Kunz-coordinates) . Let S be a numerical semigroup with multiplicity m . If Ap(
S, m ) = { w = 0 , w , . . . , w m − } , with w i congruent with i modulo m , the Kunz-coordinates vector of S is thevector x ∈ Z m − with components x i = w i − im for i = 1 , . . . , m − .We say that x ∈ Z m − is a Kunz-coordinates vector (or Kunz-coordinates, for short) if there existsa numerical semigroup whose Kunz-coordinates vector is x . From the Kunz-coordinates we can reconstruct the Ap´ery set. If x ∈ Z m − is the Kunz-coordinatesvector of S , Ap( S, m ) = { mx i + i : i = 1 , . . . , m − } ∪ { } . Consequently, S can be reconstructed fromits Kunz-coordinates.The Kunz-coordinates vectors have been implicitly used in [21] and [26] to characterize numericalsemigroups with fixed multiplicity, and used in [2] to count numerical semigroups with a given genus.Furthermore, if S is a numerical semigroup with multiplicity m and x ∈ Z m − are its Kunz-coordinates, from Selmer’s formulas, it is easy to compute its genus and its Frobenius number asfollows: • g( S ) = m − X i =1 x i . • F( S ) = max i { mx i + i } − m . (Clearly, if the maximum is reached in the i -th component,F( S ) ≡ i (mod m ))The following result that appears in [26] allows us to manipulate numerical semigroups with multi-plicity m as integer points inside a polyhedron. Theorem 14 (Theorem 11 in [26]) . Each numerical semigroup is one-to-one identified with its Kunz-coordinates.
V. BLANCO AND J. PUERTO
Furthermore, the Kunz-coordinates vectors of the set of numerical semigroups with multiplicity m is the set of solutions of the following system of diophantine inequalities: x i > for all i ∈ { , . . . , m − } , x i + x j − x i + j > for all i j m − , i + j m − , x i + x j − x i + j − m > − for all i j m − , i + j > m,x i ∈ Z + for all i ∈ { , . . . , m − } . From Theorem 14 and Selmer formulas, we can identify all the numerical semigroups (in termsof their Kunz-coordinates vector) with multiplicity m , genus g and Frobenius number F with thesolutions of this system of diophantine inequalities: x i > i ∈ { , . . . , m − } , x i + x j − x i + j > i j m − i + j m − x i + x j − x i + j − m > − i j m − i + j > m , m − X i =1 x i = gF = max i { mx i + i } − m,x i ∈ Z + for all i ∈ { , . . . , m − } . From the above formulation and Corollary 5, the set of m -irreducible numerical semigroups iscompletely determined by the solutions of the following disjunctive diophantine system of inequalitiesand equations: x i > i ∈ { , . . . , m − } , x i + x j − x i + j > i j m − i + j m − x i + x j − x i + j − m > − i j m − i + j > m , m − X i =1 x i ∈ { m − , m, max i { mx i + i } − m } x i ∈ Z + for all i ∈ { , . . . , m − } . The following result characterizes the set of oversemigroups of a numerical semigroup, in term ofits Kunz-coordinates vector.
Proposition 15.
Let S be a numerical semigroup with multiplicity m and x ∈ Z m − its Kunz-coordinates. Then, the set of Kunz-coordinates vectors of oversemigroups of S with multiplicity m is: { x ′ ∈ Z m − : x ′ is a Kunz-coordinates vector and x ′ ≤ x } where ≤ denotes the componentwise order in Z m − .Proof. Let S ′ ∈ O m ( S ), and Ap( S ′ , m ) = { , w ′ , . . . , w ′ m − } . Let Ap( S, m ) = { , w , . . . , w m − } .The i th element in the Ap´ery set is characterized of being the minimum element in the semigroupthat is congruent with i modulo m . Thus, w ′ i ≤ w i for all i = 1 , . . . , m −
1, since S ⊆ S ′ . Then x ′ i = w ′ i − im ≤ w i − im = x i for all i = 1 , . . . , m −
1. Hence, x ′ ≤ x . (cid:3) Therefore, the oversemigroups of S can be identified with the “ undercoordinates ” of its Kunz-coordinates.For the ease of presentation, we identify a numerical semigroup with multiplicity m with an integervector with m − S is a numerical semigroupand x ∈ Z m − is its Kunz-coordinates vector, we denote: • m( x ) = m( S ) = m (Multiplicity of x ). NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 7 • F( x ) = F( S ) (Frobenius number). • G( x ) = G( S ) = { n ∈ Z : mx n (mod m ) + n (mod m ) > n } (Gaps of x ). • g( x ) = g( S ) (Genus of x ). • SG( x ) = SG( S ) (Special Gaps of x ). • SG m ( x ) = SG m ( S ) (Special Gaps greater than m of x ). • U m ( x ) = { x ′ ∈ Z m − : x ′ is a Kunz-coordinates vector and x ′ ≤ x } (Undercoordinates of x ).Observe that if x is the Kunz-coordinates vector of S , x ′ ∈ U m ( x ) is univocally identified withan element S ′ ∈ O m ( S ) (Proposition 15). • Ap( x ) = Ap( S, m ) = { } ∪ { mx i + i : i = 1 , . . . , m − } (Ap´ery set of x ).Note that all the above indices and sets can be computed by using only the Kunz-coordinates vectorof the semigroup.As assumed above, we consider that S = { , m, →} . In terms of the Kunz-coordinates, it is equiva-lent to say that x = (1 , . . . , ∈ Z m − (or m − X i =1 x i ≥ m ).By Corollary 5 we say that a Kunz-coordinates vector, x ∈ Z m − is m -irreducible if g( x ) ∈ { m, m − , l F( x )+12 m } . Furthermore, we say that x is irreducible if g( x ) = l F( x )+12 m . Hence, every irreducibleKunz-coordinates vector in Z m − is m -irreducible, but the converse is not true in general.We also say that a set of Kunz-coordinates vectors, D = { x , . . . , x k } ⊆ Z m − , is a decompo-sition of x ∈ Z m − into m -irreducible Kunz-coordinates vectors if the semigroups associated withthe elements in D give a decomposition into m -irreducible numerical semigroups of the semigroupidentified with x . Equivalently, by Proposition 10, D is a decomposition of x ∈ Z m − into m -irreducible Kunz-coordinates vectors if x i is an m -irreducible Kunz-coordinates vector and SG m ( x ) =SG m ( x ) ∩ (cid:0) G( x ) ∪ · · · ∪ G( x k ) (cid:1) .Then, a minimal decomposition x ∈ Z m − into m -irreducible Kunz-coordinates is a decompositioninto m -irreducible Kunz-coordinates, D = { x , . . . , x k } ⊆ Z m − , with minimum cardinality.We define I m ( x ) = { x ′ ∈ U m ( x ) : x ′ is m -irreducible and a m -irreducible Kunz-coordinatesvector x ∗ ∈ U m ( x ) such that x ∗ ≥ x ′ }I m ( x ) is one-to-one identified with I m ( S ).4. m -irreducible Kunz-coordinates vectors In this section we give necessary and sufficient conditions for a undercoordinate of a Kunz-coordinatesvector to be m -irreducible.Let x ∈ Z m − be a Kunz-coordinates vector. By the above definition, a Kunz-coordinates vector x ′ ∈ U m ( x ) if and only if there exists y ∈ Z m − such that x ′ + y = x .By applying Theorem 14 to x ′ = x − y , y ∈ Z m − must verify the following inequalities: y i x i − i ∈ { , . . . , m − } , y i + y j − y i + j x i + x j − x i + j for all 1 i j m − i + j m − y i + y j − y i + j x i + x j − x i + j + 1 for all 1 i j m − i + j > m Actually, if we are searching for those x ′ = x − y that are identified with a set of m -irreducibleundercoordinates decomposing x , we can restrict, by Corollary 5, to consider those with genus m , V. BLANCO AND J. PUERTO m − (cid:24) F( x ) + 12 (cid:25) . Therefore, y must be a solution of the following system: y i x i − i ∈ { , . . . , m − } , y i + y j − y i + j x i + x j − x i + j for all 1 i j m − i + j m − y i + y j − y i + j x i + x j − x i + j + 1 for all 1 i j m − i + j > m, (P m ( x )) m − X i =1 y i ∈ M ( x, y ) , (1) y ∈ Z m − . where M ( x, y ) = { m − X i =1 x i − m, m − X i =1 x i − m + 1 , m − X i =1 x i − (cid:24) max i { m ( x i − y i ) + i } − m + 12 (cid:25) } .Recall that the Kunz-coordinates vector (1 , . . . , ∈ Z m − is not considered because it correspondsto S = { , m, →} that is m -irreducible, and then, its minimal decomposition is itself (Remark 8).Clearly, these coordinates are the unique solution of the above system when constraint (1) is m − X i =1 y i = m − X i =1 x i − m .In the next subsections we analyze the remaining two cases for the disjunctive constraint (1).4.1. m -irreducible undercoordinates that are irreducible. Let x ∈ Z m − be a Kunz-coordinatesvector. First, we want to find those m -irreducible undercoordinates of x that are also irreducible. Then,in system (P m ( x )), equation (1) is(2) m − X i =1 y i = m − X i =1 x i − (cid:24) max i { m ( x i − y i ) + i } − m + 12 (cid:25) . Denote now by H mk ( x ) = { y ∈ R m − : max i { m ( x i − y i ) + i } = m ( x k − y k ) + k } , and by P mk ( x ) =P( x ) ∩ H mk ( x ) for all k = 1 , . . . , m −
1. Note that H mk ( x ) is the hyperplane in R m − where the Frobeniusnumber of x − y is reached in the kth component (recall that F( x ) = max { mx i + i } − m ), that is,F( x − y ) = m ( x k − y k ) + k − m .With this assumptions, P mk ( x ) can be described by the following system of inequalities: y i x i − i ∈ { , . . . , m − } , y i + y j − y i + j x i + x j − x i + j for all 1 i j m − i + j m − y i + y j − y i + j x i + x j − x i + j + 1 for all 1 i j m − i + j > m ,(P mk ( x )) m − X i =1 y i = m − X i =1 x i − (cid:24) m ( x k − y k ) + k − m + 12 (cid:25) ,y ∈ Z m − . Or equivalently (using that z ≤ ⌈ z ⌉ < z + 1 for any z ∈ R ) by a linear system of inequalities as: NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 9 y i x i − i ∈ { , . . . , m − } , y i + y j − y i + j x i + x j − x i + j for all 1 i j m − i + j m − y i + y j − y i + j x i + x j − x i + j + 1 for all 1 i j m − i + j > m ,(P mk ( x )) 2 m − X i =1 y i − my k > m − X i =1 x i − mx k − k + m − , m − X i =1 y i − my k m − X i =1 x i − mx k − k + m − ,y ∈ Z m − . m -irreducible undercoordinates with genus m . In what follows, we describe the secondtype of m -irreducible undercoordinates of S , those with genus m .Denote by HG m ( x ) = { y ∈ R m − : m − X i =1 y i = m − X i =1 x i − m } and P mm ( x ) = P m ( x ) ∩ HG m ( x ). This setis described by the following system of diophantine inequalities: y i x i − i ∈ { , . . . , m − } , y i + y j − y i + j x i + x j − x i + j for all 1 i j m − i + j m − y i + y j − y i + j x i + x j − x i + j + 1 for all 1 i j m − i + j > m (P mm ( x )) m − X i =1 y i = m − X i =1 x i − m (3) y ∈ Z m − . The solutions of system (P mm ( x )) are easily identified by the few possible choices for the solutions ofequation (3) (the integer vector x − y ∈ Z m − has positive coordinates and the sum of them must be m ). Actually, the entire set of solutions of (P mm ( x )) is: { x − (1 , . . . , − e j : x j > , j = 1 , . . . , m − } ⊆ Z m − where e j is the j th unit vector in Z m − .Then, the set of m -irreducible undercoordinates of x with genus m is given by the set { (1 , . . . , j : x j > , j = 1 , . . . , m − } ⊆ Z m − .5. Decomposing into m -irreducible numerical semigroups In the section above we characterize the m -irreducible undercoordinates of a Kunz-coordinatesvector x ∈ Z m − . In what follows, we use these characterizations to find a decomposition of x into m -irreducible Kunz-coordinates vectors. We first give some decompositions that are not minimal ingeneral by enumerating the whole set of solutions of the systems (P mk ( x )) and (P mm ( x )). After thatwe provide a multiobjective integer programming model to obtain the set of minimal elements in I m ( x ). We prove that this model is equivalent to enumerate the entire set of optimal solutions ofsome single-objective integer programming problems. Thus, a minimal decomposition can be obtainedfrom the former set of solutions by solving a set covering problem. Finally, we propose a heuristicmethodology based on the abovementioned exact approach to obtain a (minimal) decomposition of x into m -irreducible Kunz-coordinates vectors.As a consequence of Corollary 5 and the comments above we obtain the following result that stateshow to get a decomposition into m -irreducible Kunz-coordinates vectors by solving several systems ofdiophantine inequalities. Proposition 16.
Let x ∈ Z m − be a Kunz-coordinates vector. Any decomposition of x into m -irreducible Kunz-coordinates vectors is given by some elements in the form x − y where y belongs tothe union of the solutions of the systems P m ( x ) , . . . , P mm − ( x ) and P mm ( x ) . Remark 17.
Note that the whole set of solutions of P m ( x ) , . . . , P mm − ( x ) and P mm ( x ) gives a decompo-sition into m -irreducible numerical semigroups of the semigroup S identified with x . It is the maximaldecomposition since it has the maximum possible number of m -irreducible Kunz-coordinates involved,all the m -irreducible undercoordinates of x . In the following we give a methodology to compute minimal decompositions. The main idea is tochoose, adequately, solutions of the systems P m ( x ) , . . . , P mm − ( x ) and P mm ( x ).The first step to select decompositions that are minimal with respect to the inclusion ordering is tofind the minimal elements in the set of m -irreducible undercoordinates of a Kunz-coordinates vector x . This fact can be formulated as a multiobjective integer programming problem as stated in thefollowing result. Theorem 18.
Let x ∈ Z m − be a Kunz-coordinates vector. The Kunz-coordinates vectors of theelements in I m ( x ) are in the form x − ˆ y where ˆ y is a nondominated solution of any of the followingmultiobjective integer linear programming problems. (MIP mk ( x )) v − min ( y , . . . , y m − ) s.t. y ∈ P mk ( x ) , for k = 1 , . . . , m − , m. Proof.
Let x ′ be an element in I m ( x ). Then, x ′ = x − y ′ for some y ∈ Z m − . If k = F( x ′ ) (mod m ),then, F( x ′ ) = mx ′ k + k − m . Since x ′ is an m -irreducible undercoordinate of x with the above Frobeniusnumber, either y ′ ∈ P mk ( x ) (if F( x ′ ) > m ) or y ′ ∈ P mm ( x ) (if F( x ′ ) < m ). Suppose that there is anondominated solution, ˆ y , of MIP mk ( x ) (resp. MIP mm ( x )) dominating y ′ . Then, we can find ˆ x = x − ˆ y ,with ˆ y nondominated solution of MIP mk ( x ) (resp. MIP mm ( x )) such that ˆ y ≤ y ′ and ˆ y = y ′ . Then, ˆ x ≥ x ′ and x ′ = ˆ x , and consequently, we have found an m -irreducible maximal Kunz-coordinates in I m ( x )such that ˆ x ≥ x ′ and x ′ = ˆ x , contradicting the maximality of x ′ . (cid:3) Note that, Γ, the union of the nondominated solutions of MIP m ( x ), . . . , MIP mm ( x ) contains I m ( x ),but it may contain nondominated solutions of MIP mk ( x ) that dominate some nondominated solution ofMIP mj ( x ), if k = j . Thus, Γ may contain coordinates vectors that dominate one another what wouldlead to non minimal decompositions into m -irreducible Kunz-coordinates vectors.The key to get minimal decompositions into m -irreducible Kunz-coordinates follows by applyingLemma 11. Therefore, we need to address the question about how to compute SG m ( x ). Algorithm1 shows the way of computing the special gaps greater than the multiplicity of a Kunz- coordinatesvector. This algorithm is based in the following result. There, k( n ) = n mod m stands for thenonnegative integer remainder of dividing n by m , i.e., k( n ) = n mod m . Theorem 19.
Let x ∈ Z m − be a Kunz-coordinates vector and m < h ∈ N . Then, h ∈ SG m ( x ) if andonly if h = ( x k( h ) −
1) + k( h ) and such that x k( h ) + x j > x k(k( h )+ j ) − γ k( h ) ,j for all j = 1 , . . . , m with k( h )+ j = m and h ≥ mx k(2 h ) +k(2 h ) ; and where γ ij = (cid:26) if i + j > m otherwise for all i, j = 1 , . . . , m − .Proof. The elements in SG m ( x ) are those elements fulfilling the following conditions (see [6]): • h = w i − m , where w i ∈ Ap( x ), for some i = 1 , . . . , m − • w i − w j Ap( x ) for all w j ∈ Ap( x ), w j = w i . • h ≥ w k(2 h ) By the identification of Kunz-coordinates vectors and the elements in the Ap´ery set, the first conditionsare translated in h = mx i + i − m = m ( x i −
1) + i . The second set of conditions consist of checkingfor each j = i if w i − w j = mx i + i − mx j − j
6∈ { } ∪ { mx k + k : k = 1 , . . . m − } . Note thatif mx i + i − mx j − j = mx k + k for some k , then, k( k ) = k( i − j ), so, if w i − w j is an element inAp( x ) the unique possible choice is w k( i − j ) . Now, if i > j , then k( i − j ) = i − j , and the conditionis the same as checking if mx i + i − mx j − j = mx i − j + i − j , equivalently, if x i + x i − j = x j . Since x is a Kunz-coordinates vector, by Theorem 14, x i + x i − j ≥ x j , so checking that those elements are NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 11 different is the same that x i + x i − j > x j . Clearly, by changing indices, it is that x i + x j > x i + j + γ ij (in this case i + j > m ). The case when i < j is analogous but taking into account that in that casek( i − j ) = i − j + m .The third conditions is direct from the algorithm given in [6] to compute SG m ( x ). (cid:3) The above theorem is used to compute the set SG m ( x ) for any Kunz-coordinates vector x ∈ Z m − as shown in Algorithm 1. Algorithm 1:
Computing the special gaps greater than the multiplicity of a Kunz-coordinatesvector.
Input : A Kunz-coordinates vector x ∈ Z m − .Compute M = { m ( x i −
1) + i : x i + x j > x i + j , for all j with i + j < m } and M = { m ( x i −
1) + i : x i + x j > x i + j − m − , for all j with i + j > m } . Output : SG m ( x ) = { z ∈ M ∩ M : z > m and 2 z ≥ m x k(2 z ) + k(2 z ) } .Note that the complexity of Algorithm 1 is O ( m ).From Algorithm 1 and the Kunz-coordinates vector of a numerical semigroup, we obtain the follow-ing useful result. Proposition 20.
Let x ∈ Z m − be a Kunz-coordinates vector, y ∈ Z m − and h ∈ SG m ( x ) . If x − y is a undercoordinate of x , then, h ∈ G( x − y ) if and only if y k( h ) = 0 . Furthermore, F( x − y ) is theunique element in { h ∈ SG m ( x ) : k( h ) = max { i ∈ { , . . . , m − } : y i = 0 }} .Proof. Since h ∈ SG m ( x ), by Algorithm 1, h = m ( x k( h ) −
1) + k( h ).If h ∈ G( x − y ) then, m ( x k( h ) − y k( h ) ) + k( h ) ≥ h + 1 = m ( x k( h ) −
1) + k( h ) + 1, that is, y k( h ) ≤ m − m <
1, and then y k( h ) = 0 because y i ≥ i = 1 , . . . , m − y k( h ) = 0, then, m ( x k( h ) − y k( h ) ) + k( h ) = mx k( h ) + k( h ) ≥ h + 1 since h is an specialgap of x , and then, in particular, a gap of x . Thus, h ∈ G( x − y ). (cid:3) By Proposition 10, for each h ∈ SG m ( x ) we are looking among our solution, y , for one that holds h ∈ G( x − y ). This is equivalent, by Proposition 20, to search for those with y k( h ) = 0. Then, from allthe minimal m -irreducible numerical oversemigroups of S , we only need for the minimal decomposition,those that do not contain the special gaps of S . The following result even shrink further this search. Lemma 21.
Let x ∈ Z m − be a Kunz-coordinates vector and h ∈ SG m ( x ) . Then, every nondominatedsolution of (MIP mk ( x )) , y , has y k( h ) = 0 , and then, F( x − y ) = h . Moreover, y is the solution with theminimum sum of its coordinates (length).Proof. By Algorithm 1, h = mx k( h ) + k( h ) − m . Furthermore, h ∈ G( x ′ ) for some x ′ = x − y in thedecomposition, so m ( x k( h ) − y k( h ) ) + k( h ) ≥ h + 1 = mx k( h ) + k( h ) − m + 1. Then, y k( h ) ≤ − m < y k( h ) = 0.Then, we have a feasible solution of MIP mk ( x ) with y k( h ) = 0. Therefore, for each nondominatedsolution, ˆ y , dominating y , i.e., ˆ y ≤ y and y = ˆ y . The former implies that ˆ y k( h ) = 0.Furthermore, F( x − ˆ y ) = m ( x k − y k ) + k − m = mx k + k − m = h .Since any feasible solution, y ′ , of (MIP mk ( x )) must hold m − X i =1 y ′ i = m − X i =1 x i − (cid:24) m ( x k − y ′ k ) + k − m + 12 (cid:25) ,then, m − X i =1 y ′ i > m − X i =1 x i − (cid:24) mx k + k − m + 12 (cid:25) = m − X i =1 y i , so y has minimum length, and no one else hasthis length. (cid:3) By the above result we know that, if we fix a special gap, h , a nondominated solution of MIP mk ( x )with minimum length can be computed by fixing the value of y k( h ) . Then, moving through all the special gaps in SG m ( x ) and fixing each one of them in MIP mk ( x ), we can obtain at least SG m ( x )nondominated solutions giving a decomposition of x into m -irreducible Kunz-coordinates.Therefore, an upper bound on the number of elements in any decomposition is the number of specialgaps greater than the multiplicity of the semigroup. Thus, for each problem P mk ( x ) we can add theconstraint requiring that h is a gap of the Kunz-coordinates vector, for each h ∈ SG m ( x ), i.e., y k( h ) = 0.Then, for each h ∈ SG m ( x ) and k ∈ { , . . . , m } we need to solve the following multiobjective problem:(MIP m ( x, h )) v − min ( y , . . . , y m − ) s.t. y k( h ) = 0 y ∈ P mk ( x ) Remark 22.
By Lemma 21, it is enough to search for those m -irreducible Kunz-coordinates withFrobenius numbers in SG m ( x ) . If h ∈ SG m ( x ) , this constraint is added as max i { m ( x i − y i )+ i }− m = h ,or equivalently as y k( h ) = 0 .Note that any solution of MIP m ( x, h ) is a numerical semigroup with Frobenius number congruentwith k modulo m . Since h ≡ k ( h ) (mod m ) , h ∈ SG m ( x ) , and the solutions are minimal, if onesolution has Frobenius smaller than h , then h is not in the set of gaps of those Kunz-coordinates.Then, this element irrelevant for the decomposition, since there must exist some other semigroup sothat h belongs to it. Hence, we can simplify further the decomposition process considering only single-objective integerproblems rather than multiobjective ones. The following result states this fact.
Theorem 23.
Let x be a Kunz-coordinates vector. Then, the elements in a minimal decomposition of x into m -irreducible Kunz-coordinates must belong to the union of the set of optimal solutions of thefollowing problems: (IP m ( x, h )) min m − X i =1 y i s.t. y ∈ P m k( h ) ( x ) y k ( h ) = 0 if h > m or (IP mm ( x, h )) min m − X i =1 y i s.t. y k ( h ) = x k ( h ) − ,y ∈ P mm ( x ) if h < m , for each h ∈ SG m ( x ) .Proof. Problems (MIP m ( x, h )) for any k = 1 , . . . , m − m ( x, h )) are multiobjective programswith a full dimension domination cone (see [34]). In that case, all the solutions are supported (can beobtained by solving scalars problems, or equivalently, the solutions are in the facets of the convex hullof the integer feasible region) . In our case, when fixing the special gap h , we are only interested inone of those nondominated solutions since all of them has h among its gaps, so they are irrelevant forthe minimal decomposition.Furthermore, since the improvement cone, apart of being complete, is the cone generated bye , . . . , e m − , and then, the solutions of (IP m ( x, h )) and (IP mm ( x, h )) are nondominated. Actually, weare looking for solutions with the minimum difference of gaps with x , so minimizing X i y i , and then, itis enough for our purpose to minimize the length of y as formulated in (IP m ( x, h )) and (IP mm ( x, h )). (cid:3) NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 13 Note that if (IP mm ( x, h )) is feasible, it has a unique feasible solution y = x − − e k( h ) . Furthermore,this problem is feasible if and only if k( h ) = h − m since in that case h = 2 m + k( h ) − m , the Frobeniusnumber.Actually, in this case, if (IP m ( x, h )) has also a solution, y , it must be also the solution of (IP mm ( x, h )).It is stated in the following theorem. Theorem 24.
Let x ∈ Z m − be a Kunz-coordinates vector, h ∈ SG m ( x ) and y and y optimalsolutions of problems (IP m ( x, h )) and (IP mm ( x, h )) , respectively. Then, y = y .Proof. We have two m -irreducible undercoordinates of x , x = x − y and x = x − y . x is anirreducible Kunz-coordinates vector with Frobenius number h . x is a Kunz-coordinates vector withFrobenius number h and genus m . Since the irreducible Kunz-coordinates are those with maximalgenus when fixing the Frobenius number and the maximum genus in this case is m , the genus of x isalso m , since in both problems we are minimizing the length of y . (cid:3) The following result states that when solving (IP m ( x, h )), the optimal value is known. Lemma 25.
Let y be an optimal solution of (IP m ( x, h )) . Then, X i y i = m − X i =1 x i − (cid:24) h + 12 (cid:25) Proof.
It follows directly form the satisfaction of constraint (2) and by Lemma 21. (cid:3)
Let x ∈ Z m − be a Kunz-coordinates vector. Once a decomposition is chosen, to select a minimaldecomposition, we use a set covering formulation, to choose, among the overall set of minimal m -irreducible undercoordinates of x , a minimal number of elements for the decomposition.Let SG m ( x ) = { h , . . . , h s } and D i = { x i , . . . , x i pi } be the set of the maximal Kunz-coordinatesvectors of m -irreducible undercoordinates of x when fixing the special gap h i (optimal solutions ofIP m ( x, h i )), for i = 1 , . . . , s . We denote by D = D ∪· · ·∪ D s the set of m -irreducible Kunz-coordinatesvectors candidates to be involved in the minimal decomposition of x .We consider the following set of decision variables z ij = (cid:26) x ij is selected for the minimal decomposition,0 otherwise . for i = 1 , . . . , m − j = 1 , . . . , i p i .We formulate the problem of selecting a minimal number of m -irreducible undercoordinates vectorsof x that decompose x into m -irreducible Kunz-coordinates as:(SC m ( D )) min s X i =1 p i X j =1 z ij s.t. X i,j/mx ij k( h ) +k( h ) ≥ h +1 z ij ≥ , ∀ h ∈ SG m ( x ) . The covering constraint assures that for each special gap of x there is an element in { x i , . . . , x ip , . . . , x s , . . . , x sp s } such that h is a gap of its corresponding semigroup. Minimizing theoverall sum we find the minimum number of Kunz-coordinates fulfilling this requirement. Note thatwhen solving (SC m ( D )) at most one element in D i is choosen for each i = 1 , . . . , s .Finally, if a numerical semigroup S with multiplicity m is given to be decomposed into m -irreduciblenumerical semigroups, we can, by identifying it with its Kunz-coordinates, give a procedure to computesuch a decomposition. This process is described in Algorithm 2. In that implementation we alsoconsider two trivial cases: (1) when the number of special gaps greater than the multiplicity is 1,being then the semigroup m -irreducible; and (2) when the number of this special gaps is 2, where thedecomposition is given by both solutions of the two unique integer programming problems, and nodiscarding process is needed. Algorithm 2:
Decomposition into m -irreducible numerical semigroups. Input : A numerical semigroup S with multiplicity m .Compute the Kunz-coordinates vector of S : x ∈ Z m − . (Computing the Ap´ery set.)D = {} .Compute SG m ( x ). if m ( x ) = 1 then DmIR = { x } elsefor h i ∈ SG m ( x ) doif h i < m then Set D := D ∪ { + e k( h ) } . elsefor each optimal solution of (IP m ( x, h )) , ˆ y i do Set D := D ∪ { x − ˆ y i } .Let D = { x , . . . , x i , . . . , x s , . . . , x si s } .Let z ∗ be an optimal solution of (SC m ( D )).Set DmIR = { x ij ∈ D : z ∗ ij = 1 } Output : DmIRNS = {h{ m } ∪ { mx ′ i + i : i = 1 , . . . , m − }i : x ′ ∈ DmIR } .As a consequence of all the above comments and results we state the correctness of our approach. Theorem 26.
Algorithm 2 computes, exactly, a minimal decomposition into m -irreducible Kunz-coordinates vector of a Kunz-coordinates vector x ∈ Z m − . Furthermore, the entire set of solutions (SC m ( D )) characterizes the entire set of minimal decompositions. Algorithm 2 is able to compute a minimal decomposition of a Kunz-coordinates vector, x ∈ Z m − ,by enumerating the whole set of optimal solutions of (IP m ( x, h )). However, this task is not easysince, mainly, it consists of enumerating the set of solutions of a diophantine system of inequalities,which is hard to compute (see [ ? ]). In what follows we propose an approximate approach to obtain a“short” decomposition into m -irreducibles by choosing an optimal solution of (IP m ( x, h )) instead ofenumerating all of them. One may choose any of them, but we can also slightly modify the integerprogramming model to obtain a good solution.We consider the following set of decision variables: w i = (cid:26) h i ∈ G( x − y ),0 otherwise . for i = 1 , . . . , n , and SG m ( x ) = { h , . . . , h n } .For a fixed h ∈ SG m ( x ), w i = 1 represents that h i is covered by the solution x − y , and then, thatcan be discarded to obtain a minimal decomposition.Then, to be sure that we maximize the number of elements in the previous decomposition that canbe discarded, we formulate the problem as:(IP mk ( x, h )) max m ( x ) X i =1 w i s.t. y ∈ P m k( h ) ( x ) y k ( h ) = 0 m ( x k(ˆ h i ) − y k(ˆ h i ) )+k(ˆ h i ) − ˆ h i − M (1 − w i ) ≥ h i ∈ SG m ( x )where M >> NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 15 Observe that the b ig- M constraint m ( x k(ˆ h i ) − y k(ˆ h i ) ) + k(ˆ h i ) − ˆ h i − M (1 − w i ) ≥ h G( x − y ) (equivalently m ( x k(ˆ h i ) − y k(ˆ h i ) ) + k(ˆ h i ) < ˆ h i + 1, then, w i = 0. Otherwise, w i could be0 or 1, but since we are maximizing, w i = 1.The optimal value of this integer problem is then the number of numerical semigroups in thedecomposition that can be discarded with this choice.A pseudocode of the proposed approximated scheme for obtaining a “short” decomposition of aKunz-coordinates vector x ∈ Z m − into m -irreducible Kunz-coordinates vectors by solving (IP mk ( x, h ))is shown in Algorithm 3. Algorithm 3:
Decomposition into m -irreducible numerical semigroups. Input : A numerical semigroup S with multiplicity m .Compute the Kunz-coordinates vector of S : x ∈ Z m − . (Computing the Ap´ery set.)D = {} .Compute SG m ( x ). if m ( x ) = 1 then DmIR = { x } elsefor h i ∈ SG m ( x ) doif h i < m then Set D := D ∪ { + e k( h ) } . else Let ˆ y be an optimal solution of (IP m ( x, h )). Set D := D ∪ { x − ˆ y } .Let D = { x , . . . , x s } . if m ( x ) = 2 then DmIR = D else
Select a minimal decomposition from D. Let z ∗ be an optimal solution of (SC m ( D )).Set DmIR = { x j ∈ D : z ∗ j = 1 } Output : DmIRNS = {h{ m } ∪ { mx ′ i + i : i = 1 , . . . , m − }i : x ′ ∈ DmIR } .When running Algorithm 3 we obtain an optimal solution of the problem, and then moving throughall the special gaps we obtain a decomposition into m -irreducible Kunz-coordinates. With the followingexample we show how algorithms 2 and 3 run for a given numerical semigroup. Example 27.
Let S = h , , , i . Its Kunz-coordinates vector is x = (2 , , , and SG m ( S ) = { , , } .First, we solve one integer problem for each special gap: • h = 6 : Since h < × , the integer problem to solve is P ( x, and then D = { x =(2 , , , } . • h = 13 : In this case h > × and h ≡ , so the integer problem in this caseis P ( x, . The whole set of optimal solutions is { (1 , , , , (0 , , , } , so D = { x =(2 , , , , x = (1 , , , } . • Finally, for h = 19 : clearly h > × and h ≡ , so the problem is now P ( x, .The set of optimal solutions is { (1 , , , , (0 , , , } , and then D = { x = (1 , , , , x =(2 , , , } .The above five Kunz-coordinates vectors give a decomposition in oversemigroups of S . To obtain aminimal decomposition we must solve the associated set covering problem. Solving SC ( D ) we obtain that z = z = 1 and all other variables are set to zero, being then theminimal decomposition given by x and x , i.e., a minimal decomposition into -irreducible Kunz-coordinates is given by { (2 , , , , (1 , , , } . Translating to numerical semigroups: S = h , , , , i ∩ h , , , , i When solving (IP mk ( x, h )) , we obtain the same decomposition. However, the decomposition obtained with Algorithm 3 may be not minimal. The following exampleillustrates this fact.
Example 28.
Let S = h , , , , , , , i be a numerical semigroup with multiplicity .Its Kunz-coordinates vector is x = (4 , , , , , , , , , , and SG ( x ) = { , , , , , } .Then, integer problems must be solved: IP ( x, , IP ( x, , IP ( x, , IP ( x, , IP ( x, and IP ( x, . By solving these problems with Xpress-Mosel 7.0 [17] we obtain the following opti-mal solutions x − y ∈ { (1 , , , , , , , , , , , (1 , , , , , , , , , , , (1 , , , , , , , , , , , (2 , , , , , , , , , , , (1 , , , , , , , , , , , (4 , , , , , , , , , , } The translations of the above coordinates in terms of numerical semigroups are {h , , , , , , , , , , , i , h , , , , , , , , , , , i , h , , , , , , , , , i , h , , , , , , , , , , , i , h , , , , , , , , , , , i , h , , , , , , , , , , , i} Now, by solving problem (SC m ( D )) , h , , , , , , , , , , , i is discarded. Then,the decomposition using our methodology is given by five -irreducible numerical semigroups: S = h , , , , , , , , , , , i ∩ h , , , , , , , , , i∩h , , , , , , , , , , , i ∩ h , , , , , , , , , , , i∩h , , , , , , , , , i However, this decomposition is not minimal since S = h , , , , , , , , , , , i ∩h , , , , , , , , , , , i ∩ h , , , , , , , , , , , i∩h , , , , , , , , , , , i is a decomposition into m -irreducible numerical semigroupsusing a smaller number of terms. The situation of Example 28 is due to the fact that among the whole set of optimal solutions of(IP m ( x, h )), Algorithm 3 chooses a particular one, but depending of that choice more or less elementscan be discarded from that decomposition to obtain the minimal one. To avoid this fact, we needto consider a compact model that connects all the possible elements in the decomposition and thatselects, among all of them, the smallest number of solutions to decompose a Kunz-coordinates vector.6. A compact model for minimally decomposing into m -irreducible Kunz-coordinatesvectors In the section above we describe an exact and a heuristic procedure to compute a minimal de-composition of a Kunz-coordinates vector x ∈ Z m − into m -irreducible Kunz-coordinates. To obtainsolutions by using that exact procedure we need to enumerate the solutions of a knapsack type dio-phantine equation included in the Kunz polytope. Once we have those solutions, a set covering problemmust be solved to obtain a minimal decomposition. By using that model, the complete enumerationcannot be avoided since by choosing one solution, one may obtain non-minimal decompositions whensolving the set covering model (see Example 28). We present here a compact model to decomposeany Kunz-coordinates vector, x ∈ Z m − , merging in a single integer programming problem all thesubproblems considered in the previous section to ensure minimal decompositions. Moreover, this ap-proach will allow us to prove a polynomiality result for the problem of decomposing into m -irreduciblenumerical semigoups.Let SG m ( x ) = { h , . . . , h s } .We consider the following families of decision variables for the new model: • y li ∈ Z + , such that x − y l is a m -irreducible undercoordinate of x with Frobenius number h l ,for all l = 1 , . . . , s and i = 1 , . . . , m − NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 17 • w l ∈ { , } , representing if x − y h l is chosen (1) or not (0) for a minimal decomposition into m -irreducible coordinates of x , for all l = 1 , . . . , s . • z lk ∈ { , } , that measures if h k is a gap of x − y l (1) or not (0), for all l, k = 1 , . . . , s . Notethat h k ∈ G( x − y l ) if and only if y lk ( h k ) = 0.Then, the proposed model, CIP m ( x ), is described as follows:(CIP m ( x )) min s X l =1 w l s.t. y li ≤ x i − i = 1 , . . . , m − l = 1 , . . . , s ,(4) y li + y lj − y li + j ≤ x i + x j − x i + j if i + j < m , forall l = 1 , . . . , s ,(5) y li + y lj − y li + j − m ≤ x i + x j − x i + j − m + 1 if i + j > m , forall l = 1 , . . . , s ,(6) m − X i =1 y li = ( m − X i =1 x i − (cid:24) h l + 12 (cid:25) ) w l forall l = 1 , . . . , s with h l > m ,(7) m − X i =1 y li = ( m − X i =1 x i − m ) w l forall l = 1 , . . . , s with h l < m ,(8) y lk ( h l ) = 0 forall l = 1 , . . . , s ,(9) X l z lk ( h k ) ≥ k = 1 , . . . , s ,(10) z l k( h k ) ≥ − y l k( h k ) − M (1 − w l ) forall l, k = 1 , . . . , s, (11) y lk ( h k ) ≤ M (1 − z lk ( h k ) ) forall k = 1 , . . . , s ,(12) z lk ( h k ) ≤ w l forall l, k = 1 , . . . , s ,(13) y li ∈ Z + , forall i = 1 , . . . , m − l = 1 , . . . , s ,(14) w l ∈ { , } , forall l = 1 , . . . , s ,(15) z lj ∈ { , } forall l = 1 , . . . , s , j = 1 , . . . , m − M ≥ max { x k ( h l ) : l = 1 , . . . , s } .The components of any optimal solutions, y ∗ , of the above problem in the set { y ∗ l : y ∗ l = 0 , l =1 , . . . , s } = { y ∗ l , . . . , y ∗ l p } give a minimal decomposition of x into m -irreducible Kunz-coordinatesvectors as { x − y ∗ l j : j = 1 , . . . , s } . Note also that F( x − y ∗ l j ) = h l i .Constraints (4)-(6) assure that x − y l is a undercoordinate of x . (7) and (8) give conditions relatedto the genus and the Frobenius number of those Kunz-coordinates vectors (Corollary 5) associated tothe choice of y l ( w l = 1). Constraint (9) assures that h l is a gap of x − y l , and (10) that there is atleast one element in the decomposition having h l among its gaps. Constraints (11)-(13) control thatthe variables z lk are well-defined. (14)-(16) are the integrality and binary constraints for the variables.The optimal value of (CIP m ( x )) gives the number of Kunz-coordinates involved in a minimal de-composition of x into m -irreducible Kunz-coordinates vectors.The solution of (CIP m ( x )) gives exactly a minimal decomposition of x into m -irreducible Kunz-coordinates (or m -irreducible numerical semigroups). However, it is harder to solve than the problemsin Algorithm 3 since it has much more variables (by using Algorithm 3, we need to solve at most m − m − m − m ( x )) has 2( m − + ( m −
1) integer/binary variables). In the computational experiments (seeSection 6) we have noticed that the solutions when running Algorithm 3 are not far from minimalityand it is faster than running Algorithm 2 or solving (CIP m ( x )). Remark 29 ( m -symmetry and m -pseudosymmetry) . In [6] it is also defined the notion of m -symmetryand m -pseudosymmetry of a numerical semigroup with multiplicity m , extending the previous notionsof symmetry and pseudosymmetry (see [32] ). A numerical semigroup, S , with multiplicity m is m -symmetric if S is m -irreducible and F( S ) is odd. On the other hand, S is m -pseudosymmetric if S is m -irreducible and F( S ) is even.Rosales and Branco analyzed in [27] and [28] those numerical semigroups that can be decomposedinto symmetric numerical semigroups (in this case the semigroup is called ISY-semigroup). Anotherinteresting application of this methodology is to compute a decomposition of S into m -symmetric nu-merical semigroups (following the notation in [28] , S is an ISYM-semigroup). This follows by fixing in (CIP m ( x )) that the m -irreducible numerical oversemigroups of S associated to even special gaps do notappear in the decomposition ( y li = 0 for all i = 1 , . . . , m − if l is even). Thus, the m -irreducible nu-merical semigroups which Frobenius numbers are each one of the odd special gaps must cover the wholeset of gaps. If this problem is feasible, its solution gives a minimal decomposition into m -symmetricnumerical semigroups. However, in this case we cannot ensured that it is always possible to decomposeinto m -symmetric numerical semigroups (for instance, a numerical semigroup with even Frobeniusnumber is not decomposable in this way). Then, if problem (CIP m ( x )) is infeasible, the semigroupcannot be expressed as an intersection of m -symmetric numerical semigroups.In addition, [28] analyzes the set of ISYG-semigroups (those that can be expressed as an intersectionof symmetric semigroups with the same Frobenius number). We could introduce the notion of ISYGM-semigroups (those that can be expressed as an intersection of symmetric numerical semigroups with thesame Frobenius number and multiplicity). This case can be also handled with our approach by fixingthe Frobenius number of the semigroup in (CIP m ( x )) .An similar methodology can be applied to compute a decomposition into m -pseudosymmetric numer-ical semigroups. Remark 30 (Computational Complexity) . Assume that m is fixed. (CIP m ( x )) has at most m − + ( m − variables and then, it is solvable in polynomial time [23] . It is also worth noting thatthe heuristic approach also has polynomial time overall complexity. Indeed, for each special gap of x ,one integer program is solved, IP m ( x, h ) if h > m or IP mm ( x ) if h < m . Since the number of specialgaps is bounded above by m − , the complexity of this step is polynomial for fixed multiplicity, sopolynomial. Once we have the solutions for all the special gaps, the discarding step consists of solvingthe set covering problem (SC m ( D )) with at most m − variables, so polynomial in m .On the other hand, , the algorithm proposed in [6] to decompose a numerical semigroup S withmultiplicity m into m -irreducible numerical semigroups can be rewritten as follows.Let G x = ( V, E ) be a directed graph whose set of vertices is the set of undercoordinates of x , U m ( x ) ,and ( x , x ) ∈ E if x = x − e h (mod m ) for some h ∈ SG m ( x ) . Figure 1 illustrates how this graphis built. In that figure we denote SG m ( x ) = { h , . . . , h k } and SG m ( x + e k( h ) = { h ′ , . . . , h ′ k } . Thealgorithm looks for a set o vertices { x , . . . , x n } with the properties that m ( x i ) = 1 for all i =1 , . . . , n and that any other vertex is dominated by any of the elements in the set. Furthermore, G x isa tree since it does not have circuits. In [6] , a breath first search over this tree is proposed to find thedesired set. Clearly, the worst case complexity of this method is exponential even for fixed multiplicity. x v v mmmmmmmmmmmmm ( ( QQQQQQQQQQQQQ x − e k( h w w nnnnnnnnnnnn ' ' PPPPPPPPPPPP · · · x − e k( hk ) x − e k( h − e k( h ′ · · · x − e k( h − e k( h ′ k ) Figure 1.
Sketch of G x . NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 19 Computational Experiments
In this section we present the results of some computational experiments performed to analyze theapplicability of the proposed algorithm. Our algorithm has been implemented in
XPRESS-Mosel 7.0 [17] that allows to solve the single-objective integer problems involved in the decomposition into m -irreducible numerical semigroups, by using a branch-and-bound method and nesting models by callingthe library mmjobs . The algorithms have been executed on a PC with an Intel Core 2 Quad processorat 2x 2.50 Ghz and 4 GB of RAM.The complexity of the algorithm depends of the dimension of the space (multiplicity), the size ofthe coefficients of the constraints and the number of special gaps. Then, we have randomly generatedthree different batteries of numerical semigroups with the following requirements: Battery I:
Numerical semigroups with multiplicities ranging in [0 ,
25] (divided in the five subin-tervals (0 , , , ,
20] and (20 , , Battery II:
Numerical semigroups with multiplicities ranging in [10 , , , , , , , , , Battery III:
Numerical semigroups with multiplicities ranging in [25 , , , (50 , , (75 , , (100 , , , GAP , the heuristic approach (Algorithm 3) and the compact model (CIP m ( x )). With the second set ofproblems, we check the efficiency of Algorithm 3 for solving large instances. Finally, with the third testset, we compare the difficulty of solving (CIP m ( x )) comparing to the heuristic algorithm. (Note thatthis difficulty is mainly due to the number of special gaps since it increases the number of variables.)Therefore, we generate numerical semigroups with very large multiplicities but where the number ofspecial gaps is bounded above by 30.We have used recursively the function RandomListForNS of GAP [10] until we found the list of integersdefining the semigroup with the above requirements. The implementation done for decomposing in
GAP into m -irreducible numerical semigroups is an adaptation of the function DecomposeIntoIrreducibles for decomposing into standard irreducible numerical semigroups.The results of these experiments are summarized in tables 1–3. In these tables, m indicates the rangeof the multiplicity, CMtime and
Heurtime the average times in seconds consumed by solving (CIP m ( x ))and Algorithm 3, respectively, in Xpress-Mosel , GAPtime informs on the average time consumed by
GAP for the same task, m-irred is theaverage number of semigroups involved in a minimal decomposition. The column avgap is the averagedifference between the number of numerical semigroups used in the heuristic decomposition and thenumber of numerical semigroups used in the minimal decomposition computed by solving (CIP m ( x )).Note that GAP was not able to solve any of the 10 instances when the multiplicity ranges in (20 , m CMtime Heurtime GAPtime m-irred avgap [0,5] 0.001 0.020 0.001 1.5 1.5 0(5,10] 0.003 0.054 2.3973 2.7 2.3 0(10, 15] 0.013 0.091 4.1645 4.1 3.4 0.1(15,20] 0.053 0.081 523.556 5.4 4 0(20,25] 0.046 0.089 n/a 5.7 4.4 0.1 Table 1.
Results of the computational experiments for Battery I.We have also observed that the algorithm implemented in
GAP does not ensure minimal de-compositions into m -irreducible numerical semigroups. For instance, consider the following case: m Heurtime m-irred (25,50] 0.242 11.8 7.2(50,100] 1.411 19.6 9.6(100,250] 168.272 42.4 25.4(250,500] 1318.475 86.2 47.8(500,1000] 1056.878 27.2 18.8(1000,2000] 1895.058 15.2 9.8 Table 2.
Results of the computational experiments for Battery II. m CMtime Heurtime m-irred avgap (25,50] 1.064 0.201 9.3 5.8 0.7(50,75] 6.981 0.713 13.5 7.1 1.1(75,100] 58.580 1.819 16.3 9 1(100,125] 102.999 3.428 15.1 7.1 1.6(125,150] 144.531 5.752 15.5 8.3 1.3
Table 3.
Results of the computational experiments for Battery III. S = h , , , , , , i that decomposes in GAP into six 15-irreducible numerical semigroupswhile our methodology obtains a decomposition into five 15-irreducible numerical semigroups. Thereason why
GAP fails is closely related to the same fact that prevents to ensure, in all cases, Algorithm3 to get minimal solutionsFrom our computational experiments we observe that except for the instances with m ∈ [0 , GAP spends almost the same time to compute the decompositions, our methodologysolves the problems faster than
GAP . Actually, in this battery solving the problem (CIP m ( x )) is thebest way to compute such a decomposition. This is due to the minimum computational time consumedby Xpress-Mosel to load the problems involved in Algorithm 3.Both, the exact algorithm based on solving (CIP m ( x )) and the heuristic approach are able tocompute, in reasonable CPU times, minimal decompositions into m -irreducible numerical semigroupsfor multiplicities up to 150 while the procedure implemented in GAP is not able to solve problems withmultiplicities ranging even in (20 , m ( x )) for larger multiplicities, the heuristic approach solves problems withmultiplicities up to m = 2000.The heuristic approach finds, much faster than the exact approach, a short decomposition of anumerical semigroups into m -irreducible numerical semigroups. Furthermore, the heuristic approachreaches most of the times a minimal decomposition. For instance, in the first battery of problems,the heuristic value does not coincide with the exact optimal one in only two out the fifty instances.Moreover, the third battery of instances satisfies that in 30% of the cases the minimal decompositioncoincides with the heuristic short decomposition, in 34% of the cases the difference is only one semi-group, in 30% of the cases is two semigroups, in 4% (two cases) is three and in only 2% (one instance)is four.Note that most of the computations done by using Algorithm 3 may be parallelized by solving indifferent cores each one of the problems (IP mk ( x, h )) since they are independent. This could improve theCPU times and sizes of the problems because more than 99% of the time consumed by this algorithm isto solving those problems, while just a little part of the time is spent solving the set covering problem.On the other hand, we have only implemented the proposed models in Xpress-Mosel , with the defaultbranch-and-bound method. Larger instances could be solved by applying specific more sophisticatedinteger programming algorithms to solve each one of the problems.
NTEGER PROGRAMMING AND m -IRREDUCIBILITY OF NUMERICAL SEMIGROUPS 21 Acknowledgments
This research has been partially supported by the Spanish Ministry of Science and Education grants
MTM2007- 67433-C02-01 and
MTM2010-19576-C02-01 . The first author have been also supportedby Juan de la Cierva grant
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Departamento de ´Algebra, Universidad de Granada
E-mail address : [email protected] Departamento de Estad´ıstica e Investigaci´on Operativa, Universidad de Sevilla
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