Discrete Polymatroids Research Articles

The divisor class group of a discrete polymatroid

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

Fertilitopes

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation pi is a particular set mathcal V(pi ) of integer compositions that appears in a formula for the fertility of pi , which is defined to be |s^{-1}(pi )|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that mathcal V(pi ) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of pi to be the convex hull of mathcal V(pi ), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of pi directly from pi using Bousquet-Mélou’s notion of the canonical tree of pi . As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that sum _{sigma in s^{-1}(pi )}x^{textrm{des}(sigma )+1} is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

Hodge Theory for Polymatroids

Abstract We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized Goresky–MacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard Lefschetz, and Hodge–Riemann theorems for the Chow ring. Furthermore, we provide a relative Lefschetz decomposition with respect to the deletion of an element.

Double Schubert polynomials do have saturated Newton polytopes

Abstract We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

Products of polymatroids with the strong exchange property

It was conjectured by White in 1980 that the toric ring associated to a matroid is defined by symmetric exchange relations. This conjecture was extended to discrete polymatroids by Herzog and Hibi, and they prove that the conjecture holds for polymatroids with the so called strong exchange property. In this paper we generalize their result to polymatroids that are products of polymatroids with the strong exchange property. This also extends a result by Conca on transversal polymatroids.

Design of ideal secret sharing based on new results on representable quadripartite matroids

With the rapid development of the next generation of mobile networks and communications (5G networks), group-oriented applications show the great potential ability in 5G and multipartite group communications occur frequently in a single network, where a multipartite access structure is denied to be a collection of the subsets of users who may come from different parts of the network such that only users in an authorized subset of users can use their shares to build up a group key for a secure group communication. Most existing group key establishment schemes based on a secret sharing target on building up a group key for a threshold access structure and need to compute a t-degree interpolating polynomial in order to encrypt and decrypt the secret group key. This approach is not suitable and inefficient in terms of computational complexity for multipartite group environments which need to realize the multipartite access structures. In this paper, we study the characterization of ideal access structures related to representable multipartite matroids. By employing some existing results, such as the well-known connection between ideal secret sharing and matroids, the recent discovery of ideal multipartite access structures, and the connection between multipartite matroids and discrete polymatroids, we obtain the characterization of a family of representable multipartite matroids, including a sufficient condition for an access structure to be ideal. Based on this result, we provide a complete characterization of representable quadripartite matroids, showing that these access structures related to representable quadripartite matroids are the ideal ones. From these ideal access structures, we can then construct ideal secret sharing schemes for multipartite group communications. Our results are attractive for efficient and secure multi-group-oriented applications.

Generalized Index Coding Problem and Discrete Polymatroids.

The connections between index coding and matroid theory have been well studied in the recent past. Index coding solutions were first connected to multi linear representation of matroids. For vector linear index codes, discrete polymatroids, which can be viewed as a generalization of the matroids, were used. The index coding problem has been generalized recently to accommodate receivers that demand functions of messages and possess functions of messages. In this work we explore the connections between generalized index coding and discrete polymatroids. The conditions that need to be satisfied by a representable discrete polymatroid for a generalized index coding problem to have a vector linear solution is established. From a discrete polymatroid, an index coding problem with coded side information is constructed and it is shown that if the index coding problem has a certain optimal length solution then the discrete polymatroid is representable. If the generalized index coding problem is constructed from a matroid, it is shown that the index coding problem has a binary scalar linear solution of optimal length if and only if the matroid is binary representable.

A Discrete Polymatroidal Framework for Differential Error-Correcting Index Codes

In the conventional index coding problem, the messages transmitted by the source to satisfy the demands of the receivers are error-free. However, in practice, the messages may be error prone which led to the introduction of error-correcting index codes. An error-correcting index code is an encoding scheme which enables every receiver to decode its demanded message even in the presence of certain number of transmission errors. Formally, an index code capable of correcting at most $\delta $ errors at all its receivers is defined as a $\delta $ -error-correcting index code. In this paper, we explore the connections between vector linear error-correcting index codes and discrete polymatroids. This is motivated from the connections between linear network error-correcting codes and matroids. It is shown that a vector linear error-correcting index code exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. Differential error-correcting index codes which allow different receivers to have different error-correcting capabilities are also studied. Note that the $\delta $ -error-correcting code is a special case of a differential error-correcting index code. Error correction at a subset of receivers is another special case which is discussed.

Discrete polymatroids satisfying a stronger symmetric exchange property

In this paper we introduce discrete polymatroids satisfying the one-sided strong exchange property and show that they are sortable (as a consequence their base rings are Koszul) and that they satisfy White's conjecture. Since any pruned lattice path polymatroid satisfies the one-sided strong exchange property, this result provides an alternative proof for one of the main theorems of J. Schweig in [12], where it is shown that every pruned lattice path polymatroid satisfies White's conjecture. In addition we characterize a class of such polymatroids whose base rings are Gorenstein. Finally for two classes of pruned lattice path polymatroidal ideals I and their powers we determine their depth and their associated prime ideals, and furthermore determine the least power k for which depthS/Ik and Ass(S/Ik) stabilize. It turns out that depthS/Ik stabilizes precisely when Ass(S/Ik) stabilizes in both cases.

Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids

Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding, and representable discrete polymatroids. We consider the vector linear solutions of networks over a field $\mathbb {F}_{q}$ , with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. It is well known that a scalar linear solution over $\mathbb {F}_{q}$ exists for a network if and only if the network is matroidal with respect to a matroid representable over $\mathbb {F}_{q}$ . We define a discrete polymatroidal network and show that a linear fractional solution over a field $\mathbb {F}_{q}$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb {F}_{q}$ . An algorithm to construct the networks starting from certain class of discrete polymatroids is provided. Every representation over $\mathbb {F}_{q}$ for the discrete polymatroid, results in a linear fractional solution over $\mathbb {F}_{q}$ for the constructed network. Next, we consider the index coding problem, which involves a sender, which generates a set of messages $X=\{x_{1},x_{2},\dotso x_{k}\}$ , and a set of receivers $\mathcal {R}$ , which demand messages. A receiver $R \in \mathcal {R}$ is specified by the tuple $(x,H)$ , where $x \in X$ is the message demanded by $R$ and $H \subseteq X \!\setminus \! \{x\}$ is the side information possessed by $R$ . We first show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid, satisfying certain conditions, which are determined by the index coding problem considered. Rouayheb et al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid. The multi-linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. We generalize the result of Rouayheb et al. , by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.

New Results on Ideal Multipartite Secret Sharing and its Applications to Group Communications

With the rapid development of various group-oriented services, multipartite group communications occur frequently in a single network, where a multipartite access structure is defined to be a collection of the subsets of users who may come from different parts of the network such that only users in an authorized subset of users can use their shares to build up a group key for a secure group communication. Most existing group key establishment schemes based on a secret sharing target on building up a group key for a threshold access structure, and need to compute a $$t$$t-degree interpolating polynomial in order to encrypt and decrypt the secret group key. This approach is not suitable and inefficient in terms of computational complexity for multipartite group environments which need to realize the multipartite access structures. In 1991, Brickell et al. proved that an ideal access structure is induced by a matroid and furthermore, an access structure is ideal if it is induced by a representable matroid. In this paper, we study the characterization of representable matroids. By using the connection between ideal secret sharing and matroids and, in particular, the recent results on ideal multipartite access structures and the connection between multipartite matroids and discrete polymatroids, we introduce a new concept on $$R$$R-tuple, which is determined by the rank function of the associated discrete polymatroid. Using this new concept, we come up a new and simple sufficient condition for a multipartite matroid to be representable (in fact, every matroid and every access structure are multipartite). In other words, we have developed a sufficient condition for an access structure to be ideal. These new results can be applied to establish multipartite group keys efficiently in secure group communications.

New Results on Multipartite Secret Sharing Matroids

In a secret-sharing scheme, a secret value is distributed among a set of participants by giving each participant a share. The requirement is that only predefined subsets of participants can recover the secret from their shares. The family of the predefined authorized subsets is called the access structure. An access structure is ideal if there exists a secret-sharing scheme realizing it in which the shares have optimal length, that is, in which the shares are taken from the same domain as the secrets. Brickell and Davenport (J. of Cryptology, 1991) proved that ideal access structures are induced by matroids. Subsequently, access structures induced by matroids and properties of matroids have received a lot of attention. Since every matroid is multipartite and has the corresponding discrete polymatroid, in this paper, by dealing with the bases of discrete polymatroids, we obtain a necessary condition for a set of vectors to be the bases of a discrete polymatroid, which implies a necessary condition for a multipartite access structure to be ideal. Concretely, we describe the properties of the bases of bipartite, tripartite and quadripartite matroids respectively, which show the necessary conditions for multipartite access structures induced by bipartite, tripartite and quadripartite matroids to be ideal accordingly. Our results give new contributions to the characterization of ideal access structures, which is one of the main open problems in secret sharing.

Multipartite matroids and secret sharing

In a secret-sharing scheme, a secret value is distributed among a set of participants by giving each participant a share. The requirement is that only predefined subsets of participants can recover the secret from their shares. The family of the predefined authorized subsets is called the access structure. An access structure is ideal if there exists a secret-sharing scheme realizing it in which the shares have optimal length, that is, in which the shares are taken from the same domain as the secrets. Brickell and Davenport proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have received a lot of attention. Seymour gave the first example of an access structure induced by a matroid namely the Vamos matroid, that is non-ideal. Since every matroid is multipartite and has the associated discrete polymatroid, in this paper, by dealing with the rank functions of discrete polymatroids, we obtain a sufficient condition for a multipartite access structure to be ideal. Furthermore, we give a new proof that all access structures related to bipartite and tripartite matroids coincide with the ideal ones. Our results give new contributions to the open problem, that is, which matroids induce ideal access structures.

On the h-Vector of a Lattice Path Matroid

Stanley has conjectured that the h-vector of a matroid complex is a pure M-vector. We prove a strengthening of this conjecture for lattice path matroids by constructing a corresponding family of discrete polymatroids.

Symmetric and quasi-symmetric functions associated to polymatroids

To every subspace arrangement X we will associate symmetric functions ℘[X] and ℋ[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant ℋ[X] specializes to the Tutte polynomial \({\mathcal{T}}[\mathbf{X}]\) . Billera, Jia and Reiner recently introduced a quasi-symmetric function ℱ[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function \({\mathcal{G}}[\mathbf{X}]\) for polymatroids which has this property as well. Moreover, \({\mathcal{G}}[\mathbf{X}]\) specializes to ℘[X], ℋ[X], \({\mathcal{T}}[\mathbf{X}]\) and ℱ[X].

Equidimensional and Unmixed Ideals of Veronese Type

This article was motivated by a problem left by Herzog and Hibi, namely, to classify all unmixed polymatroidal ideals. In the particular case of polymatroidal ideals corresponding to discrete polymatroids of Veronese type, i.e., ideals of Veronese type, we give a complete description of the associated prime ideals, and then we show that such an ideal is unmixed if and only if it is Cohen–Macaulay. We also give for this type of ideals equivalent characterizations for being equidimensional.

Ideals of fiber type and polymatroids

In the first half of this paper, we complement the theory on discrete polymatroids. More precisely, (i) we prove that a discrete polymatroid satisfying the strong exchange property is, up to an affinity, of Veronese type; (ii) we classify all uniform matroids which are level; (iii) we introduce the concept of ideals of fiber type and show that all polymatroidal ideals are of fiber type. On the other hand, in the latter half of this paper, we generalize the result proved by Stefan Blum that the defining ideal of the Rees ring of a base sortable matroid possesses a quadratic Grobner basis. For this purpose we introduce the concept of ``$l$-exchange property'' and show that a Grobner basis of the defining ideal of the Rees ring of an ideal $I$ can be determined and that $I$ is of fiber type if $I$ satisfies the $l$-exchange property. Ideals satisfying the $l$-exchange property include strongly stable ideals, polymatroid ideals of base sortable discrete polymatroids, ideals of Segre-Veronese type and certain ideals related to classical root systems.

Discrete Polymatroids

The discrete polymatroid is a multiset analogue of the matroid. Based on the polyhedral theory on integral polymatroids developed in late 1960's and in early 1970's, in the present paper the combinatorics and algebra on discrete polymatroids will be studied.