Consequences of the packing problem
Hrishikesh Bodas, Benjamin Drabkin, Caleb Fong, Su Jin, Justin Kim, Wenxuan Li, Alexandra Seceleanu, Tingting Tang, Brendan Williams
aa r X i v : . [ m a t h . A C ] J a n CONSEQUENCES OF THE PACKING PROBLEM
HRISHIKESH BODAS, BENJAMIN DRABKIN, CALEB FONG, SU JIN, JUSTIN KIM,WENXUAN LI, ALEXANDRA SECELEANU, TINGTING TANG, BRENDAN WILLIAMS
Abstract.
We study several consequences of the packing problem, a conjecture fromcombinatorial optimization, using algebraic invariants of square-free monomial ideals.While the packing problem is currently unresolved, we successfully settle the validityof its consequences. Our work prompts additional questions and conjectures, whichare presented together with their motivation. Introduction
The packing problem introduced by Conforti and Cornu´ejols [CC90] is a conjec-ture originating from combinatorial optimization in the context of max-flow min-cutproperties. It has been brought into commutative algebra through the inspiring paper[GVV05] of Gitler, Valencia and Villarreal. A comprehensive account of this problemfrom an algebraic perspective can be found in the monograph [Vil15, § + I has a unique irredundant decomposition of I intoprime ideals which takes the form I = P ∩ · · · ∩ P s with P i = ( x j , . . . , x j sj ) for j = 1 , . . . , s . Furthermore, one defines the height of I , ht( I ) to be the minimum of Mathematics Subject Classification.
Primary 13C70, 13F55, 05E40; Secondary 05C65, 05C15.
Key words and phrases. monomial ideals, symbolic powers, linear programming, packing problem,Newton polyhedron, symbolic polyhedron.The second author was supported by the NSF RTG grant in algebra and combinatoricsat the University of Minnesota DMS–1745638. The seventh author was supported by NSFDMS–1601024. This work was completed in the framework of the 2020 Polymath program https://geometrynyc.wixsite.com/polymathreu . the number of minimal generators of the prime components P i . Based on the abovedecomposition, for each positive integer m one defines the m -th symbolic power of I tobe the monomial ideal I ( m ) = P m ∩ · · · ∩ P ms . Symbolic power ideals are important in algebraic geometry where they encode poly-nomial functions vanishing to high order on a given algebraic variety. They are alsorelevant in combinatorics. For example, if I is the edge ideal of a (hyper)graph, the m -th symbolic power of I encodes the m -covers of the (hyper)graph.We single out a class of squarefree monomial ideals which is important to this project. Definition 1.1.
A a square-free monomial ideal I is K¨onig if there is a set of pairwisecoprime monomials in I of cardinality ht( I ).The ideal I has the packing property if every ideal obtained from I by setting a(possibly empty) subset of the variables equal to 0 and a disjoint (possibly empty)subset of the variables equal to 1 is K¨onig.The terminology K¨onig is best explained by the connection to K¨onig’s theorem onbipartite graphs; see the discussion following Theorem 2.10 and the terminology packedis explained by the relationship to edge packings in hypergraphs. See Conjecture 2.15for a combinatorial formulation of the packing problem which clarifies this perspective. Conjecture 1.2 (The packing problem – [GVV05]) . The symbolic and ordinary powersof a squarefree monomial ideal I coincide, i.e. I ( m ) = I m for all positive integers m , ifand only if I has the packing property. While the direct implication of Conjecture 1.2 is known to hold, cf. [DDSG +
17, p. 26],the converse implication is at the time of this writing a long standing conjecture. Forthe case of graphs, Conjecture 2.15 holds by [GVV05]. In fact for the edge ideal I ( G ) ofa graph G the following are equivalent: I ( G ) has the packing property, I ( G ) is K¨onig, G is bipartite, I ( G ) ( m ) = I ( G ) m for all m ≥
1. Previous work on the packing problemincludes [CGM98, GRV05, HM10, TT11, MV12, MnNnb18, AB19].Our work establishes that three consequences of the converse implication in thepacking problem hold. The first consequence gives a numerical shadow of the equalityof the ordinary and symbolic powers of an ideal in the form of an equality between theinitial degree and the Waldschmidt constant. The initial degree α ( I ) of a homogeneousideal I is the least degree of a nonzero element of I . The Waldschmidt constant of I can be viewed as an asymptotic initial degree for the family of symbolic powers of I .This invariant is defined as b α ( I ) = lim m →∞ α ( I ( m ) ) m ; see Definition 3.7 for details.The second consequence gives a shadow of the equality of the ordinary and symbolicpowers of an ideal in convex geometric terms. In detail, there are two convex bodiesthat can be associated to the families of ordinary and symbolic powers of a monomialideal I respectively, see [CDF + I , N P ( I ), and the symbolic polyhedron of I , SP ( I ) cf. Definition 3.1 and Definition 3.3.We show that these two polyhedra are equal for ideals which have the packing property.Equivalently linear programs having these two convex bodies as feasible sets have thesame solutions. ONSEQUENCES OF THE PACKING PROBLEM 3
Our main results on consequences of the packing problem are summarized below:
Theorem (Theorem 4.1, Theorem 4.2, Theorem 4.3) . If I is a square-free monomialideal which satisfies the packing property then there are equalities α ( I ) = b α ( I ) and N P ( I ) = SP ( I ) , as predicted by the packing problem. Moreover the optimal solutionfor any linear program with feasible set N P ( I ) coincides with the optimal solution forthe linear program with the same objective function and feasible set SP ( I ) . We also study the relationship between the packing property and Alexander duality,with the following conclusion.
Theorem (Corollary 5.12) . Let I be an equidimensional square-free monomial ideals I such that I ∨ is also equidimensional. Then I and I ∨ satisfy the packing propertysimultaneously, that is, I satisfies the packing property if and only if I ∨ does. Our paper is organized as follows: section 2 provides a dictionary between square-free monomial ideals and hypergraphs, presents several combinatorial optimizationinvariants of hypergraphs and restates the packing problem in combinatorial language.Section 3 introduces several convex bodies and combinatorial optimization invariantsfor monomial ideals. Three consequences of the packing problem are introduced andproven in section 4 as Theorem 4.1, Theorem 4.2, Theorem 4.3. In section 5, we discussthe irreversibility of the consequences of the packing problem formulated in this paper,single out the class of uniform hypergraphs as a possible candidate for which theconverses of our results may apply, and prove Corollary 5.12 regarding the relationshipbetween the packing property and Alexander duality.2.
Square-free monomial ideals and hypergraphs
In this section we present the fundamental dictionary relating square-free monomialideals to hypergraphs, also known as clutters. We supplement this dictionary by inter-preting some hypergraph and ideal theoretic invariants by means of linear optimization.An excellent reference for this theory is [Vil15].We denote by N the set of non negative integers and by [ n ] the set { , . . . , n } .2.1. Square-free monomial ideals as edge ideals of hypergraphs.
An ideal ofthe polynomial ring R = K [ x , . . . , x n ] with coefficients in a field K is a monomial ideal if it is generated by monomials. It is a square-free monomial ideal if it is generatedby square-free monomials, i.e., every generator has the form x i · · · x i t with i j ∈ [ n ]. Asquare-free monomial ideal I has a unique irredundant decomposition into prime idealswhich takes the form I = P ∩ · · · ∩ P s with P i = ( x j , . . . , x j sj ) for j ∈ [ s ] . The prime ideals P j appearing in this decomposition are called the associated primes of I and form a set denoted Ass( I ). The height of a square-free monomial ideal I isthe least number of variables needed to generate any of its associated primes, i.e.,ht( I ) = min P ∈ Ass( I ) ht( P ) = min j ∈ [ s ] s j . POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM A hypergraph is an ordered pair H = ( V, E ) where V = { x , . . . , x n } is the set of vertices , and E consists of subsets of V such that if e i ⊆ e j , then e i = e j . The elementsof E are called edges . When the cardinality of each edge is | e i | = 2, H is a graph .There is a bijective correspondence between hypergraphs H on n vertices and square-free monomial ideals of R = K [ x , . . . , x n ] by means of the following construction. Definition 2.1.
Given any hypergraph H = ( V, E ), one can associate to H a square-free monomial ideal I ( H ) called the edge ideal of H . Precisely, we define I ( H ) = ( x i x i · · · x i t | { i , i , . . . , i t } ∈ E ) . The correspondence between hypergraphs H and square-free monomial ideals I ( H )extends to a dictionary relating combinatorial invariants of H to algebraic invariantsof I ( H ). For example, the associated primes of I ( H ) are related to the maximalindependent sets and minimal vertex covers of the hypergraph H . We say that A ⊆ V is an independent set of H if e A whenever e ∈ E . It is maximal if it is maximalwith respect to inclusion among all independent sets of H .A subset U ⊆ V is a vertex cover or transversal of a hypergraph if e ∩ U = ∅ whenever e ∈ E . A vertex cover is minimal if it is so with respect to containment. A minimum vertex cover is a vertex cover of smallest cardinality. Note that a minimumvertex cover is minimal, but the converse need not be true. The cardinality of anyminimum vertex cover for a hypergraph H is denoted τ ( H ) and termed the transversalnumber of the hypergraph H .The following lemma gives a formal description of the relationship between associatedprimes of I ( H ) and minimal vertex covers and maximal independent sets of H . Lemma 2.2.
Suppose that H = ( V, E ) is a hypergraph with E = ∅ and let I = I ( H ) .Let I = P ∩ · · · ∩ P s be the irredundant prime decomposition of I , and set P ′ i = { x j | x j P i } for i ∈ [ s ] . Then, identifying the set of generators for each of theseideals with the set of corresponding vertices in V , yields (1) P , . . . , P s are the minimal vertex covers of H (2) P ′ , . . . , P ′ s are the maximal independent sets of H , (3) ht( I ( H )) = τ ( H ) .Proof. The first statement is proven in [Vil15, Lemma 6.3.37]. The last statementfollows from the first and the definitions for height and τ ( H ). For the second statement,any P ′ i is a maximal independent set if and only if V \ P i is a minimal vertex cover.We now use the first claim to finish the proof. (cid:3) Linear optimization invariants of hypergraphs.
The transversal number of ahypergraph introduced above can be described as the solution of an integer optimizationproblem. To formulate the problem, we introduce incidence matrices.
Definition 2.3.
The incidence matrix of the hypergraph H = ( V, E ) with V = { v , . . . , v n } and E = { e , . . . , e t } is the n × t matrix given by(2.1) B i,j = ( v i ∈ e j v i e j . ONSEQUENCES OF THE PACKING PROBLEM 5
The following lemma utilizes the notation z = (cid:2) z · · · z n (cid:3) T for a column vector in R n , for the zero vector in R n , and for the vector in R n with all entries equal to 1.Moreover, inequalities between vectors are understood componentwise. Lemma 2.4.
The transversal number τ ( H ) of a hypergraph H is the optimum valueof the following integer program (2.2) minimize z + · · · + z n subject to B T z ≥ and z ∈ N n .Proof. A vector z ∈ Z n satisfies the inequality B T z ≥ if and only if for each edge e i ∈ E there is some j ∈ [ n ] such that v j ∈ e i and z j ≥ v ( z ) = { v j | z j ≥ } is a vertex cover for H . The linear program (2.2) seeks tominimize the cardinality of the vertex cover v ( z ), in accordance to the definition of thetransversal number. (cid:3) An edge packing or matching of a hypergraph H = ( V, E ) is a subset of disjointedges, i.e. D ⊆ E such that no two elements of D share a vertex. Since the edges of H are in bijection with the minimal monomial generators of the edge ideal I ( H ) and twoedges are disjoint if and only if the monomials representing them in I ( H ) are coprime,we have the following description for edge packing in algebraic terms: Remark . Edge packings of a hypergraph H are in bijection with subsets of theminimal monomial generators of I ( H ) in which the elements are pairwise coprime.Maximal and maximum edge packings are defined to be the edge packings that aremaximal with respect to containment and to cardinality, respectively. The size of amaximum edge packing is called the packing number of H , denoted π ( H ). The packingnumber of a hypergraph is also the solution to an integer optimization problem withconstraints given by the incidence matrix (2.1), which we now describe. Lemma 2.6.
The packing number of a hypergraph π ( H ) is the optimum value of thefollowing integer linear program (2.3) maximize y + · · · + y t subject to B y ≤ and y ∈ N t .Proof. A vector y ∈ N t satisfies the inequality B y ≤ if and only if for each vertex v i ∈ V there at most one j ∈ [ n ] such that v i ∈ e j and y j ≥ e ( y ) = { e i | y i ≥ } is a packing for H . The linear program (2.3) seeks to maximizethe cardinality of the packing e ( y ), in accordance to the definition of the packingnumber. (cid:3) Solving integer optimization problems is much harder than solving linear optimiza-tion problems in R n because the simplex algorithm solves the latter problem efficiently,while there are no efficient algorithms to solve the former. Therefore a standard prac-tice is to consider the real relaxation of an integer program. The relaxations of theinteger programs in (2.2) and (2.3) are described below. This follows a well estab-lished trend to study fractional invariants of combinatorial structures; see [SU11] foran overview of this method. POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM
Definition 2.7.
Define the fractional transversal number τ f ( H ), of a hypergraph H to be the optimum solution for the following linear program(2.4) minimize z + · · · + z n subject to B T z ≥ and z ≥ .The feasible set of the linear program above is termed the set covering polyhedron in[Vil15]. This is polyhedron is defined as follows(2.5) Q ( H ) = { z ∈ R n | B T z ≥ , z ≥ } . If Q ( H ) is an integer polyhedron, meaning that its vertices have integer coordinates,then the hypergraph H is called Fulkersonian or ideal . Any Fulkersonian hypergraph H satisfies τ f ( H ) = τ ( H ). This class of hypergraphs is analyzed from an algebraicperspective in [Tru06, EVY06] . Definition 2.8.
Define the fractional packing number of a hypergraph H , π f ( H ), tobe the optimum solution for the relaxation of packing problem, namely(2.6) maximize y + · · · + y t subject to B y ≤ and y ≥ .An important tool in linear programming is linear program duality. This is exempli-fied by the linear programs (2.4) and (2.6), which are dual to each other. A core aspectof linear optimization is that (real) dual linear programs have the same optimum value,hence there is an equality π f ( H ) = τ f ( H ). In fact, based on linear programing duality,Lemma 2.4, and Lemma 2.6 we deduce the following inequalities(2.7) π ( H ) ≤ π f ( H ) = τ f ( H ) ≤ τ ( H ) . It is natural to ask under what circumstaces there is equality among the four in-variants involved in equation (2.7). In combinatorial optimization one considers moregenerally pairs of dual integer programs with arbitrary objective function. When thesepairs have equal optimum values the hypergraph is said to satisfy the max-flow min-cutproperty .We note that the equality π ( H ) = τ ( H ) is equivalent to asking for the edge ideal I ( H ) to be K¨onig cf. Definition 1.1. Remark . A hypergraph H satisfies the equality π ( H ) = τ ( H ) if and only if I ( H ) isK¨onig. Indeed, by Remark 2.5, π ( H ) is the cardinality of the largest set of pairwise co-prime monomials among the generators of I , whereas by Lemma 2.2 τ ( H ) = ht( I ( H )).The following celebrated theorem of K¨onig and Egerv´ary provides a context in whichthe equality π ( H ) = τ ( H ) is achieved for graphs. Together with the preceding remark,this shows that edge ideals of bipartite hypergraphs are K¨onig. Theorem 2.10 (K¨onig, Egerv´ary) . In any bipartite graph G , the number of edges ina maximum matching equals the number of vertices in a minimum vertex cover, i.e τ ( G ) = π ( G ) . ONSEQUENCES OF THE PACKING PROBLEM 7
The packing problem as a combinatorial optimization problem.
In thissection we reformulate the packing problem Conjecture 2.15 in terms of the linearoptimization invariants of hypergraphs introduced above.Remark 2.9 suggests the following definition:
Definition 2.11.
A hypergraph H is K¨onig if it satisfies the equality τ ( H ) = π ( H ).Following the convention in [FHM13], we define two operations on hypergraphs toget smaller hypergraphs. Definition 2.12. A deletion in a hypergraph is the removal of a vertex v from thevertex set and the removal of any edges that contain it from the edge set. A contraction in a hypergraph is the removal of a vertex from the vertex set and from any edges thatcontain it.A minor of a hypergraph H = ( V, E ) is a hypergraph obtained through a sequenceof deletions and contractions. More precisely, it is a hypergraph h = ( V \ ( V ′ ∪ V ”) , { e \ V ′′ | e ∈ E, e ∩ V ′ = ∅} obtained by fixing disjoint (possibly empty) sets V ′ , V ′′ ⊆ V , deleting all vertices in V ′ and contracting all vertices in V ′′ .We translate Definition 2.12 into algebraic language as follows. Lemma 2.13. If h is a minor of H then I ( h ) is obtained from I ( H ) by setting thevariables corresponding to v ∈ V ′ equal to 0 and the variables corresponding to v ′′ ∈ V ′′ equal to 1.Proof. This follows from the description of the edge set of the minor h in Definition 2.12. (cid:3) Based on Definition 2.11 and Lemma 2.13, we can translate the packing property ofsquare-free monomial ideals Definition 1.1 into an equivalent definition for hypergraphs.
Definition 2.14.
A hypergraph H is said to have the packing property if every minor h of H is K¨onig, that is, satisfies τ ( h ) = π ( h ).Finally, we can restate the packing problem in combinatorial language: Conjecture 2.15 (The packing problem - hypergraph version) . A hypergraph H sat-isfies I ( H ) ( m ) = I ( H ) m for all m ∈ N if and only if H has the packing property. Linear optimization invariants of monomial ideals
In this section we introduce some algebraic invariants of monomial ideals which canbe realized as solutions of linear optimization problems and we expand upon theirrelationship to the combinatorial optimization invariants from the previous section.
POLYMATH 2020, MONOMIALS, CONVEX BODIES, AND OPTIMIZATION TEAM
Convex bodies associated to monomial ideals.
For a homogeneous ideal I the initial degree, denoted α ( I ), is the least degree of a non zero element of I . Weshow that the initial degree of a monomial ideal can be expressed as the solution of alinear program. For this, we first define the feasible region of the program. Definition 3.1.
The
Newton polyhedron of a monomial ideal I is the convex hull ofthe exponent vectors of all monomials in I , namely N P ( I ) = convex hull { ( a , . . . , a n ) ∈ N n | x a · · · x a n n ∈ I } . Newton polyhedra of monomial ideals I are integer (or lattice) polyhedra, meaningthat their vertices have integer coordinates. Indeed, the vertices of N P ( I ) are theexponent vectors for a subset of the minimal generators of I .With this notation, the initial degree of a monomial ideal I can be expressed as thesolution of a linear program as follows. Lemma 3.2. If I is a monomial ideal then the initial degree α ( I ) is the solution ofthe following linear program (3.1) minimize a + · · · + a n subject to a = ( a , . . . , a n ) ∈ N P ( I ) .Proof. This follows because the optimal solution is attained at a vertex of
N P ( I )and, as remarked above, the vertices of N P ( I ) correspond to a subset of the minimalgenerators of I . Thus the optimum value of the linear program (3.3) corresponds to aminimal generator of I of least degree. (cid:3) While the vertices of the Newton polyhedron are easy to understand, the dual de-scription in terms of bounding inequalities is often difficult to come by. Below wedescribe a different polyhedron obtained from a square-free monomial ideal which hasthe advantage that its bounding inequalities can be read off the prime decompositionof the ideal.
Definition 3.3.
The symbolic polyhedron of a square-free monomial ideal I with primedecomposition I = P ∩ · · · ∩ P s such that P j = ( x j , · · · , x j sj ) for j ∈ [ s ] is defined tobe the intersection of the Newton polyhedra of the prime components SP ( I ) = N P ( P ) ∩ · · · ∩ N P ( P s )Equivalently, y = ( y , . . . , y n ) ∈ R n is a point in SP ( I ) if and only if it satisfies ( y j + · · · + y j sj ≥ ≤ j ≤ sy i ≥ ≤ i ≤ n. In contrast to the Newton polyhedron, the symbolic poyhedron is a rational polyhe-dron, meaning that its vertices have rational coordinates. For applications of the sym-bolic polyhedron, including relationships to combinatorics, see [BCG +
16, CEHH17].The following result elucidates the relationship between the Newton and symbolicpolyhedra of a monomial ideal.
ONSEQUENCES OF THE PACKING PROBLEM 9
Proposition 3.4.
Let I be a square-free monomial ideal. Then, there is a containment N P ( I ) ⊆ SP ( I ) and the two polyhedra have the same lattice points, that is, N P ( I ) ∩ N n = SP ( I ) ∩ N n . Proof.
Let I be a square-free monomial ideal, with decomposition into prime monomialideals given by I = P ∩ · · · ∩ P s for some s ∈ N . The containment N P ( I ) ⊆ SP ( I )follows from the considering the containments I ⊆ P i which yield N P ( I ) ⊆ N P ( P i )for i ∈ [ s ]. Therefore we conclude N P ( I ) ⊆ T si =1 N P ( P i ) = SP ( I ).The previous containment implies N P ( I ) ∩ N n ⊆ SP ( I ) ∩ N n . Let a = ( a , . . . , a n ) ∈ SP ( I ) ∩ N n be a lattice point in SP ( I ). It follows that for all i ∈ [ s ] we have a ∈ N P ( P i ). It is well known that the lattice points in the Newton polyhedron ofa monomial ideal correspond to monomials in the integral closure of the ideal [HS06,Proposition 1.6], hence x a := x a · · · x a n n ∈ P i , where P i denotes the integral closure of P i . Since monomial prime ideals are integrally closed, we conclude that x a ∈ P i for all i ∈ [ s ], thus x a ∈ I and a ∈ N P ( I ) ∩ N n , as desired. (cid:3) We can now give an alternate description for the initial degree of a square-freemonomial ideal. To do this we need to associate a matrix to the prime decompositionof a monomial ideal.
Definition 3.5.
For a square-free monomial ideal I with prime decomposition I = P ∩· · · ∩ P s such that P j = ( x j , · · · , x j sj ) for j ∈ [ s ] we define a s × n prime decompositionmatrix with entries(3.2) A i,j = ( x j ∈ P i x j P i .In the following statement a = (cid:2) a · · · a n (cid:3) T denotes a vector in R n . Lemma 3.6. If I is a square-free monomial ideal then the initial degree α ( I ) is theoptimal solution of the following equivalent linear programs (3.3) minimize a + · · · + a n minimize a + · · · + a n subject to a ∈ SP ( I ) ∩ N n subject to A a ≥ and a ∈ N n . Proof.
That α ( I ) is the optimal solution of the leftmost linear program follows fromLemma 3.2, Proposition 3.4 and the fact that N P ( I ) is an integer polyhedron. Itremains to show the equivalence of the two linear programs in the statement. Thisreduces to showing they have the same feasible region. Comparing the bounding in-equalities for SP ( I ) provided in Definition 3.3 to the inequalities in the rightmost linearprogram defined using the prime decomposition matrix (3.2) one concludes that theycoincide. (cid:3) We now turn to the relaxation of the linear program in (3.3), which yields a fractionalversion of the initial degree. It turns out that this algebraic invariant has first appearedin the literature under a different guise, which we now recall.
Definition 3.7 ([BH10]) . The
Waldschmidt constant of a homogeneous ideal I is thevalue of the following limit b α ( I ) = lim m →∞ α ( I ( m ) ) m . It turns out that the sequence { α ( I ( m ) ) } m ∈ N is subadditive as shown by the contain-ments I ( m ) I ( m ′ ) ⊆ I ( m + m ′ ) for all m, m ′ ∈ N . Farkas’s lemma [Far02] thus applies toshow that the limit in Definition 3.7 exists and is equal to the infimum of the respectivesequence.The following theorem shows that the Waldschmidt constant of a square-free mono-mial ideal is the optimum value of the relaxation of the linear program (3.3). Theorem 3.8 ([CEHH17, Corollary 6.3]) . For a square-free monomial ideal I withprime decomposition matrix A as in (3.2) , the Waldschmidt constant b α ( I ) is the opti-mum value of the following equivalent linear programs (3.4) minimize a + · · · + a n minimize a + · · · + a n subject to a ∈ SP ( I ) subject to A a ≥ and a ≥ . Alexander duality.
To relate the algebraic invariants for monomial ideals in-troduced in section 3 to the combinatorial invariants for hypergraphs encountered insection 2.2 it is convenient to introduce the notion of Alexander duality.
Definition 3.9.
Let I be a square-free monomial ideal with prime decomposition I = P ∩ · · · ∩ P s , where P j = ( x j , · · · , x j sj ) for j ∈ [ s ]. The Alexander dual of I is thesquare-free monomial ideal I ∨ = ( x j x j · · · x j sj | j ∈ [ s ]) . If H is the hypergraph with edge ideal I = I ( H ) we define the dual hypergraph of H as the hypergraph H ∨ whose edge ideal is I ( H ) ∨ , i.e., I ( H ∨ ) = I ( H ) ∨ . In thecombinatorial optimization literature H ∨ is called the blocker of H .The importance of Alexander duality in our setting is that it interchanges the theprime decomposition matrix (3.2) and the (transpose of the) incidence matrix (2.1).The following observation arises from comparing Definition 3.5, Definition 2.3, andDefinition 3.9. Lemma 3.10. If I is a square-free monomial ideal, H is the hypergraph satisfying I = I ( H ) , A and A ∨ denote the prime decomposition matrices of I and I ∨ respectively,and B and B ∨ denote the adjacency matrices of H and H ∨ respectively, then A ∨ = B T and B ∨ = A T . This simple observation shows how the algebraic invariants of monomial ideals relateto combinatorial invariants of hypergraphs.
Corollary 3.11. If I is a square-free monomial ideal and H is the hypergraph satisfying I = I ( H ) , then the initial degree and Waldschmidt constant of I can be expressed interms of the (fractional) transversal number of the dual hypergraph H ∨ as follows: α ( I ) = τ ( H ∨ ) , b α ( I ) = τ f ( H ∨ ) and τ ( H ) = α ( I ∨ ) , τ f ( H ) = b α ( I ∨ ) . ONSEQUENCES OF THE PACKING PROBLEM 11
We now turn to convex geometric relationships between the symbolic polyhedron ofa square-free monomial ideal, and the set covering polyhedra Q ( H ) and Q ( H ∨ ) for thecorresponding hypergraph and its dual. Corollary 3.12.
Let I be a square-free monomial ideal and let H be the hypergraphsatisfying I = I ( H ) . The following are equivalent (1) the symbolic polyhedron of I is an integer polyhedron, (2) the hypergraph H is Fulkersonian, (3) the dual hypergraph H ∨ is Fulkersonian, (4) the symbolic polyhedron of I ∨ is an integer polyhedron.Proof. From Definition 3.3, equation (2.5), and the previous lemma, it follows that SP ( I ) = Q ( H ∨ ). The latter is an integer polyhedron if and only if H ∨ is Fulkersonian,establishing the equivalence of (1) and (2). The equivalence of (2) and (3) follows from[Ber89, Corollary, p. 210]. The equivalence of (3) and (4) follows from the equivalenceof (1) and (2) by duality. (cid:3) Using Lemma 2.13 one can express contraction and deletion as dual operationsthrough the lens of Alexander duality.
Lemma 3.13. If I is a square-free monomial ideal, H is the hypergraph satisfying I = I ( H ) , V ′ , V ′′ are subsets of the vertex set of H , and H ′ and H ′′ are the minors of H obtained by deleting V ′ and contracting V ′′ respectively then (1) ( I ( H ′ )) ∨ = ( I | { x ′ v =0 | v ′ ∈ V ′ } ) ∨ = I ∨ | { x ′ v =1 | v ′ ∈ V ′ } and (2) ( I ( H ′′ )) ∨ = I ∨ | { x ′′ v =0 | v ′′ ∈ V ′′ } = ( I | { x ′′ v =1 | v ′′ ∈ V ′′ } ) ∨ .In particular, ( H ′ ) ∨ is obtained from H ∨ by contracting V ′ and ( H ′′ ) ∨ is obtained from H ∨ by deleting V ′′ .Proof. Lemma 2.13 yields I ( H ′ ) = I | { x ′ v =0 | v ′ ∈ V ′ } and I ( H ′′ ) = I | { x ′′ v =1 | v ′′ ∈ V ′′ } . Suppose I = P ∩ · · · ∩ P s is the irredundant prime decomposition of I and that P i ∩ V ′′ = ∅ ifand only if i ∈ [ t ]. Set P ′ i = ( x v | x v ∈ P i \ V ′ ), m i = Q x v ∈ P i x v and m ′ i = Q x v ∈ P ′ i x v = m i | { x ′ v =1 | v ′ ∈ V ′ } . Then a prime decomposition of I ( H ′ ) is I ( H ′ ) = s \ i =1 P i | { x ′ v =0 | v ′ ∈ V ′ } = s \ i =1 P ′ i , which yields I ( H ′ ) ∨ = ( m ′ , . . . , m ′ s ) = ( m , . . . , m s ) | { x ′ v =1 | v ′ ∈ V ′ } = I ∨ | { x ′ v =1 | v ′ ∈ V ′ } . More-over, a prime decomposition of I ( H ′′ ) is I ( H ′′ ) = T ti =1 P i which yields I ( H ′′ ) ∨ = ( m , . . . , m t ) = ( m , . . . , m s ) | { x ′′ v =0 | v ′′ ∈ V ′′ } = I ∨ | { x ′′ v =0 | v ′′ ∈ V ′′ } . (cid:3) Consequences of the packing problem
We now turn our attention to establishing some consequences of the packing problem.These are recorded in Theorem 4.1, Theorem 4.2, and Theorem 4.3, which constitute the main results of this section. Their interconnections are summarized in the followingsequence of implications elaborated upon in below(4.1)
T heorem . Conjecture .
15 (packing problem)
T heorem . T heorem . Theorem 4.1. If I is a square-free monomial ideal which satisfies the packing propertythen SP ( I ) = N P ( I ) . The second is a linear optimization shadow of Conjecture 1.2.
Theorem 4.2. If I is a square-free monomial ideal which satisfies the packing propertyand f ( a ) = c a + · · · + c d a d is any linear function with c i ≥ for each i then thefollowing two linear programs have equal optimum values: (4.2) minimize c a + · · · + c d a d minimize c a + · · · + c d a d subject to a ∈ SP ( I ) subject to a ∈ N P ( I ) . The restriction c i ≥ i insures that the optimum values of the linearprograms in (4.2) are real numbers. If this is not satsified, then the optimum values ofboth programs are −∞ . This is because both the Newton and the symbolic polyhedronare closed under increasing coordinates.The third consequence is a numerical shadow of Conjecture 1.2. Theorem 4.3. If I is a square-free monomial ideal which satisfies the packing propertythen there is an equality b α ( I ) = α ( I ) . We start by showing that the validity of each of the above theorems follows from thevalidity of the packing problem Conjecture 1.2. For this purpose we recall the followingresult regarding points with rational coordinates in symbolic polyhedra.
Lemma 4.4 ([CEHH17, Proposition 6.1]) . Let I be a monomial ideal with symbolicpolyhedron SP ( I ) . For any a ∈ SP ( I ) ∩ Q d there exists a positive integer b such that x m a ∈ I ( m ) whenever m is divisible by b . Armed with this result, we are now ready to prove a general result regarding theequality of the symbolic and Newton polyhedra.
Proposition 4.5.
Let I be a monomial ideal such that I ( n ) = I n for all n ≥ . Thenthere is an equality N P ( I ) = SP ( I ) . ONSEQUENCES OF THE PACKING PROBLEM 13
Proof.
To prove the claim, it suffices to show that every vertex of SP ( I ) lies in N P ( I ).By convexity of N P ( I ) and SP ( I ), this would imply SP ( I ) ⊆ N P ( I ). Let a =( a , . . . , a d ) ∈ SP ( I ) be a vertex. Since the bounding hyperplanes of SP ( I ) are givenby equations with integer coefficients by Definition 3.3, we have that a ∈ Q n . ByLemma 4.4, we can choose some positive integer b such that x b a ∈ I ( b ) = I b . Thisimplies that b a ∈ N P ( I b ) = b · N P ( I ), which allows us to conclude that a ∈ N P ( I ) asrequired.The argument above shows that SP ( I ) ⊆ N P ( I ). Since the reverse inclusion holdsin general (see Proposition 3.4), we obtain the desired equality. (cid:3) The preceding result allows us to show Conjecture 1.2 implies Theorem 4.1.
Proposition 4.6.
Assume that the assertion of the packing problem, Conjecture 1.2,is true. Then any square-free monomial ideal I which has the packing property satisfiesthe equality N P ( I ) = SP ( I ) .Proof. Let I be a square-free monomial ideal which satisfies the packing property.Assuming that the statement of the packing problem is true, the hypothesis impliesthat the equalities I ( n ) = I n hold for all n ∈ N . By Proposition 4.5, it follows that SP ( I ) = N P ( I ). (cid:3) We next show the equivalence of Theorem 4.1 and Theorem 4.2 connecting the equal-ity of the Newton and symbolic polyhedra and the equivalence of linear programs withnonnegative coefficients for the objective function. .
Proposition 4.7.
Let I be a monomial ideal. Then there is an equality of polyhedra SP ( I ) = N P ( I ) if and only if the following linear programs have the same optimalsolution for all objective functions f ( a ) = c a + · · · + c n a n with c i ≥ for each i ∈ [ n ](4.3) minimize c a + · · · + c n a n minimize c a + · · · + c n a n subject to a ∈ SP ( I ) subject to a ∈ N P ( I ) .Proof. The forward implication is clear. We now focus on the converse.Consider the symbolic polyhedron SP ( I ) and the Newton polyhedron N P ( I ) of I and recall from Proposition 3.4 that N P ( I ) ⊆ SP ( I ). To show that N P ( I ) = SP ( I ), itsuffices to show that the vertices of SP ( I ) are contained in N P ( I ). Let p = ( p , . . . , p d )be a vertex of SP ( I ). Note that p is in fact the intersection point of n distinct boundinghyperplanes H , . . . , H n for N P ( I ). By Definition 3.3, these bounding hyperplaneshave non-negative (with entries 0 and 1) normal vectors c , . . . , c n respectively. Takingtheir arithmetic mean c = 1 n ( c + . . . + c n ) = ( c , . . . , c n ) ∈ R n ≥ , we obtain a hyperplane H p with equation c ( a − p ) + · · · + c n ( a n − p n ) = 0 whichintersects SP ( I ) only at p . This is because the equation of H p is the arithmetic meanof the equations of H , . . . , H n and for each point of SP ( I ) other than p the result ofsubstituting it into the equations of H , . . . , H n is always non-negative, with at least one positive value. Since H p ∩ SP ( I ) = { p } , the optimal value of the linear programminimize f ( a ) = c a + · · · + c d a d subject to a ∈ SP ( I )is attained at p . Since c ≥ , the hypothesis implies that f ( p ) is the optimal value ofthe linear program with objective function f and feasible set N P ( I ). Since f ( a ) > f ( p )for all points a ∈ SP ( I ) \ { p } and since N P ( I ) ⊆ SP ( I ), it follows that f ( a ) > f ( p )for all points a ∈ N P ( I ) \ { p } . We conclude that the point p must belong to N P ( I )in order for f ( p ) to be the minimum value of the second linear program in (4.3). Since p was an arbitrary vertex of SP ( I ), it follows that SP ( I ) ⊆ N P ( I ), as desired. (cid:3) The final implication needed to complete diagram (4.1) is the following.
Lemma 4.8.
Theorem 4.2 implies Theorem 4.3.Proof.
Setting c i = 1 for i ∈ [ n ] in the linear programs (4.2) yields the linear programs(3.4) and (3.3) respectively. By Lemma 3.6 and Theorem 3.8 the optimal values ofthese programs are α ( I ) and b α ( I ). Thus the conclusion of Theorem 4.2 implies that α ( I ) = b α ( I ), i.e., the conclusion of Theorem 4.3. (cid:3) Finally, we prove Theorem 4.3, Theorem 4.1, and Theorem 4.2 independent of the(not yet established) validity of the packing problem. Due to the implications indiagram (4.1), it suffices to prove the validity of Theorem 4.1, since this result impliesthe other two. Towards this end we use a celebrated result of Lehman.
Theorem 4.9 ([Leh90], [CC90, Theorem 1.8]) . If a hypergraph H has the packingproperty, then the polyhedron Q ( H ) defined in equation (2.5) is an integer polyhedron. We are now ready to prove our main results.
Proof of Theorem 4.1.
Let H and H ∨ be the hypergraphs determined by I ( H ) = I and I ( H ∨ ) = I ∨ respectively. Since I and hence H satisfy the packing property byhypothesis, Theorem 4.9 yields that the set covering polyhedron Q ( H ) is an integerpolyhedron, thus H is Fulkersonian. Corollary 3.12 now yields that I has an inte-ger symbolic polyhedron. Since the vertices of SP ( I ) are lattice points, they belongto N P ( I ) by Proposition 3.4, thus inducing a containment SP ( I ) ⊆ N P ( I ). Sincethe opposite containment always holds (see Proposition 3.4) we conclude the desiredequality SP ( I ) = N P ( I ). (cid:3) As previously remarked, the lattice points in the Newton polyhedron
N P ( I ) of anymonomial ideal I correspond to monomials in the integral closure I of the ideal I ; see[HS06, Proposition 1.6]). Thus the Newton polyhedra N P ( I ) and N P ( I ) coincide.Thus the equality SP ( I ) = N P ( I ) can be rewritten as SP ( I ) = N P ( I ) and thoughtof as capturing the equality of symbolic powers and integral closures of powers of I .We present an alternate proof of Theorem 4.1 based on this intuition. For this purposewe recall an alternate description of the symbolic polyhedron from [CDF + ONSEQUENCES OF THE PACKING PROBLEM 15
Lemma 4.10 ([CDF +
20, Corollary 3.12]) . Let I be a monomial ideal. Then the sym-bolic polyhedron of I can be described as SP ( I ) = [ m ≥ m N P ( I ( m ) ) . Alternate proof of Theorem 4.1.
As in the previous proof of this result, the hypothesisthat I satisfies the packing property implies that H is Fulkersonian. By [Tru06, The-orem 2.3] or [EVY06, Proposition 3.4] this guarantees equality of the symbolic powersand integral closures of ordinary powers, namely I ( m ) = I m for each m ≥
1. Passingto the respective convex bodies yields the identities
N P ( I ( m ) ) = N P ( I m ) = N P ( I m ) = mN P ( I )and taking the convex limit of the above family of polyhedra yields the desired equality SP ( I ) = [ m ≥ m N P ( I ( m ) ) = [ m ≥ m · mN P ( I ) = [ m ≥ N P ( I ) = N P ( I ) . (cid:3) Further questions and conjectures
In this section we consider the implications that our results have on the packingproblem. This amounts to reversing the implications in diagram (4.1). While we showbelow that in general these implications are not reversible, this line of reasoning leadsus to some related conjectures that have a bearing on the packing problem.5.1.
Uniform hypergraphs.
In Proposition 4.6 we showed that the packing problemimplies the validity of Theorem 4.1. We note that Theorem 4.1, not imply the validityof Conjecture 2.15 (the packing problem) and neither do Theorem 4.2 or Theorem 4.3,as illustrated by the following Example 5.1. This is closely related to the irreversibilityof Theorem 4.9 also demonstarted by this example. We first learned of Example 5.1from [DD20, Remark 5.4]. In combinatorial optimization the corresponding hypergraphhas gained some recognition under the name Q , see [Vil15, Example 14.2.9]. It is aforbidden minor of any hypergraph that satisfies the max-flow min-cut property. Example 5.1.
Consider the square-free monomial ideal I = ( abc, aef, cde, bdf ) ⊆ K [ a, b, c, d, e, f ]with prime decomposition I = ( a, d ) ∩ ( b, e ) ∩ ( c, f ) ∩ ( a, b, c ) ∩ ( a, e, f ) ∩ ( b, d, f ) ∩ ( c, d, e ) , which implies that ht( I ) = 2. It can be verified using a computer algebra system suchas Macaulay2 [GS] that there is an equality
N P ( I ) = SP ( I ) and consequently SP ( I )is an integer polyhedron and α ( I ) = b α ( I ).However, the ideal I does not satisfy the packing property. In particular, we seethat the ideal I itself is not K¨onig as any ht( I ) = 2 monomials in I have non-trivialcommon divisor. This ideal fails to satisfy the packing property in a minimal way, since it is only the full ideal I that is not K¨onig. All other minors obtained by settingany number of variables equal to 1 or 0 are K¨onig. Definition 5.2.
A hypergraph H is uniform if every edge of H has the same number ofvertices, equivalently if the edge ideal I ( H ) is equigenerated. An ideal is equidimesional if all its associated primes have the same height.The two notions above are related by Alexander duality: if I is the edge ideal of ahypergraph H , then I is equidimensional if and only if H ∨ is uniform and H is uniformif and only if I ∨ is equidimensional.We note that the ideal in Example 5.1 is not equidimensional. We do not knowwhether Theorem 4.1 and Conjecture 1.2 (the packing problem) are equivalent for equidimensional ideals. This motivates the following question. Question 5.3.
Is there an equidimensional square-free monomial ideal I so that there isan equality of polyhedra SP ( I ) = N P ( I ) (equivalently, SP ( I ) is an integer polyhedron),but I is not packed or I m = I ( m ) for some n ? If the answer to Question 5.3 is negative, then this means Theorem 4.1 and thepacking problem are equivalent for equidimensional ideals. Since we have provenTheorem 4.1, this would imply the validity of the packing problem for equidimensionalideals.5.2.
Partite hypergraphs.
Recall that a graph satisfies the packing property if andonly if it is bipartite. In this section we investigate notions of partite hypergraphs andtheir relationship to Waldschmidt constants of edge ideals. We make a conjecture inthis regard, which would provide a new bridge from Conjecture 2.15 to Theorem 4.3.
Notation . Let A be a set, and k be a positive integer. We denote by (cid:0) Ak (cid:1) the set of k element subsets of A , i.e. (cid:0) Ak (cid:1) = { S ⊆ A : | S | = k } . Definition 5.5.
Let H = ( V, E ) be a hypergraph and let a and b be positive integerswith a ≥ b . A function f : V → (cid:0) [ a ] b (cid:1) is called an ( a : b )– partition or ( a : b )– rainbowcoloring of H if for each edge e and for each color i , there is a vertex v ∈ e such that i ∈ f ( v ). A hypergraph H is said to be ( a : b ) –partite or ( a : b ) –rainbow colorable if ithas an ( a : b )–rainbow coloring.We say that H is a -partite or a -colorable if H has an ( a : 1)–coloring.We show below how the property of a graph of being ( a : b )–partite imposes alower bound on the Waldschmidt constant of its edge ideal. Towards this end, we firstintroduce a useful property of partite hypergraphs. Lemma 5.6.
Let H be an ( a : b ) –partite hypergraph. Then there exist disjoint minimalvertex covers C , . . . , C a such that no vertex appears in more than b of the vertex covers.Proof. Fix an ( a : b )–rainbow coloring f of H = ( V, E ), and consider the a color classesof this coloring, i.e., the sets A i = { v ∈ V | i ∈ f ( v ) } . Since f is an ( a : b )–rainbowcoloring of H , every color appears at least once in each edge, and thus the color classesare vertex covers. Thus each A i contains a minimal vertex cover C i . No vertex appears ONSEQUENCES OF THE PACKING PROBLEM 17 in more than b of the C i , since if it did, that would mean that there is a vertex withmore than b colors, which is a contradiction. (cid:3) The proof of the following result uses the description of the Waldschmidt constantas the solution of a linear optimization problem in Theorem 3.8.
Proposition 5.7.
Let H be a hypergraph with edge ideal I = I ( H ) and let a and b bepositive integers with a ≥ b . If H is ( a : b ) –partite, then the inequality b α ( I ) ≥ ab holds.Proof. Let I = P ∩ · · · ∩ P s be the irredundant prime decomposition of I . Since H is( a : b )–partite, by Lemma 5.6 there is a set of a minimal vertex covers of H such thatno vertex appears in more than b of the covers. Minimal vertex covers of H correspondto associated primes of I by Lemma 2.2. Without loss of generality let P , . . . , P a bethe primes corresponding to these minimal vertex covers. From Definition 3.3, we have SP ( I ) = N P ( P ) ∩ · · · ∩ N P ( P a ) ∩ · · · ∩ N P ( P s ) . where for P j = ( x j , . . . , x j sj ) one has N P ( P j ) = n y | y j + · · · + y j sj ≥ , y j , · · · , y j sj ≥ o . In particular, any point c = ( c , . . . , c n ) ∈ SP ( I ) satisfies the set of a inequalities: s j X k =1 c j k ≥ j ∈ [ a ] . Adding up these inequalities one obtains for α i = |{ j ∈ [ a ] | x i ∈ P j }| the equation n X i =1 α i c i ≥ a. Since no vertex appears in more than b of the a minimal vertex covers correspondingto P , . . . , P a , we have that α i ≤ b for each i , leading to the inequality n X i =1 bc i ≥ n X i =1 α i c i ≥ a. It follows that c + c + · · · + c n ≥ ab . As c was arbitrary, minimizing the sum ofcoordinates over SP ( I ) as in Theorem 3.8, we reach the conclusion b α ( I ) ≥ ab . (cid:3) We have shown in section 4 that the validity of the packing problem for an ideal I implies the equality b α ( I ) = α ( I ). In general, for arbitrary ideals I , the inequality b α ( I ) ≤ α ( I ) holds thus to have equality it suffices to show the converse inequality. Ournext conjecture is inspired by the lower bound given by Proposition 5.7. We conjecturethat in the presence of the packing property the inequality in Proposition 5.7 yields aninequality b α ( I ) ≥ α ( I ) by means of a specific partite structure on the correspondinghypergraph. Conjecture 5.8.
Let H be a hypergraph with the packing property, and let I = I ( H ) be its edge ideal. Then H is ( α ( I ) ht( I ) : ht( I )) –partite. Packing and Alexander duality.
Satisfying the packing property need not bepreserved under taking the Alexander dual. In fact the Alexander dual of an ideal I that satisfies the packing property can fail to satisfy this property, even in the casewhen I is equidimensional and hence the hypergraph corresponding to I ∨ is uniform. Example 5.9.
Consider Example 5.1 again where we introduced the ideal I = ( abc, aef, cde, bdf ) = ( a, d ) ∩ ( b, e ) ∩ ( c, f ) ∩ ( a, b, c ) ∩ ( a, e, f ) ∩ ( b, d, f ) ∩ ( c, d, e ) . Its Alexander dual is I ∨ = ( ad, be, cf, abc, aef, bdf, cde ) = ( a, b, c ) ∩ ( a, e, f ) ∩ ( c, d, e ) ∩ ( b, d, f ) . Using the computer algebra system Macaulay2 [GS] equipped with the package
Symbo-licPowers [DGSS19], one can check that I does not satisfy the packing property, while I ∨ does. Notice also that the ideal I ∨ is equidimensional, while I is not equidimensional.This example leads to the following task. Question 5.10.
Give an algebraic or combinatorial description for the class of square-free monomial ideals I such that I satisfies the packing property if and only if I ∨ does. In Corollary 5.12 we give a partial answer by showing that equidimensional ideals I which have equidimensional duals I ∨ are part of the class of ideals singled out inQuestion 5.10. Note that the family of equidimensional ideals I such that I ∨ is alsoequidimensional corresponds bijectively to uniform hypergraphs H which have a uni-form blocker H ∨ .We begin our inquiry by studying properties of uniform hypergraphs which have thepacking property. Theorem 5.11.
Let H be a uniform hypergraph which satisfies the packing propertyand let I = I ( H ) . Then (1) H has an exact cover, meaning a vertex cover which meets every edge in exactlyone vertex, (2) H is α ( I ) –partite, (3) I ∨ and H ∨ satisfy the packing property.Proof. Because H satisfies the packing property, the polyhedron Q ( H ) is integral byTheorem 4.9. A uniform hypergraph with integral set covering polyhedron has an exactcover by [GVV05], see also [Vil15, Lemma 14.4.1].We now show that H is α ( I )–partite by induction on α ( I ). If α ( I ) = 1 the claimfollows since every hypergraph is 1-partite. Otherwise, let C be an exact cover of H anddenote h the minor of H obtained by deleting the vertices in C . The monomial ideal I ( h ) satisfies the packing property and α ( I ( h )) = α ( I ) −
1, therefore h is ( α ( I ) − h to H by coloring the vertices in C with a new color, yields that H is α ( I )–partite, as desired.Let A i denote the set of vertices colored by color i ∈ [ α ( I )]. Then each set A i is a vertex cover and thus contains a minimal vertex cover C i of H . Since the sets A i are disjoint, so are the minimal vertex covers C i . Consequently, I ∨ contains a ONSEQUENCES OF THE PACKING PROBLEM 19 regular sequence of α ( I ) monomial generators m i = Q v ∈ C i x v . Since ht( I ∨ ) = α ( I ) bydefinition, it follows that I ∨ is K¨onig according to Definition 1.1.To show that I ∨ has the packing property, consider subsets V ′ , V ′′ of the verticesof H . Since H is uniform, the minor H ′ of H obtained by deleting the vertices in V ′ is a uniform hypergraph. Its edge ideal is I ( H ′ ) = I | { x ′ v =0 | v ′ ∈ V ′ } . The identity fromLemma 3.13 ( I ( H ′ )) ∨ = ( I | { x ′ v =0 | v ′ ∈ V ′ } ) ∨ = I ∨ | { x ′ v =1 | v ′ ∈ V ′ } reveals that the ideal I ∨ | { x ′ v =1 | v ′ ∈ V ′ } is K¨onig, because we have shown above that theduals of uniform hypergraphs that satisfy the packing property are K¨onig.Now consider the minor H ′′ obtained from H by contracting the vertices in V ′′ . Thisneed not be a uniform hypergraph. Its edge ideal is I ( H ′′ ) = I | { x ′′ v =1 | v ′′ ∈ V ′′ } . Let v = |{ i | C i ⊆ V ′′ }| . We claim that H ′′ is ( α ( I ) − v )–partite. Indeed, H ′′ can be colored with α ( I ) − v colorsusing the color assignments v i if v ∈ C i \ V ′′ . Since the color sets are disjoint, themonomials m i = Q v ∈ C i x v form a regular sequence in I ( H ′′ ) ∨ of length α ( I ) − v .Consider the ideal discussed in Lemma 3.13 I ′′ := ( I ( H ′′ )) ∨ = ( I | { x ′′ v =1 | v ′′ ∈ V ′′ } ) ∨ = I ∨ | { x ′′ v =0 | v ′′ ∈ V ′′ } and let ht( I ′′ ) = α ( I ( H ′′ )) = u . To establish the claim that I ′′ is K¨onig, it suffices toshow that u ≤ α ( I ) − v . (In fact this will force u = α ( I ) − v .) Note that the inequality u ≤ α ( I ) − v is equivalent to v ≤ α ( I ) − α ( I ( H ′′ )). Let m ′′ be a minimal generator of I ( H ′′ ) with deg( m ′′ ) = α ( I ( H ′′ ). Then there is a minimal generator m of I such thatdeg( m ) = α ( I ) and m/m ′′ is a product of α ( I ) − α ( I ( H ′′ )) variables corresponding tovertices in V ′′ . Since the sets C i are vertex covers, each C i ⊆ V ′′ must contain at leastone vertex corresponding to a variable dividing the monomial m/m ′′ . Since the sets C i are disjoint, this observation yields the desired conclusion v ≤ α ( I ) − α ( I ( H ′′ )). (cid:3) Corollary 5.12.
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