37 research outputs found
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 vector linear solutions of
networks over a field with possibly different message and edge
vector dimensions, which are referred to as linear fractional solutions. We
define a \textit{discrete polymatroidal} network and show that a linear
fractional solution over a field exists for a network if and
only if the network is discrete polymatroidal with respect to a discrete
polymatroid representable over An algorithm to construct
networks starting from certain class of discrete polymatroids is provided.
Every representation over for the discrete polymatroid, results
in a linear fractional solution over for the constructed
network. Next, we consider the index coding problem and 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. El Rouayheb et. al. showed
that the problem of finding a multi-linear representation for a matroid can be
reduced to finding a \textit{perfect linear index coding solution} for an index
coding problem obtained from that matroid. We generalize the result of El
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.Comment: 24 pages, 6 figures, 4 tables, some sections reorganized, Section VI
newly added, accepted for publication in IEEE Transactions on Information
Theor
Linear Fractional Network Coding and Representable Discrete Polymatroids
A linear Fractional Network Coding (FNC) solution over is a
linear network coding solution over in which the message
dimensions need not necessarily be the same and need not be the same as the
edge vector dimension. Scalar linear network coding, vector linear network
coding are special cases of linear FNC. In this paper, we establish the
connection between the existence of a linear FNC solution for a network over
and the representability over of discrete
polymatroids, which are the multi-set analogue of matroids. All previously
known results on the connection between the scalar and vector linear
solvability of networks and representations of matroids and discrete
polymatroids follow as special cases. An algorithm is provided to construct
networks which admit FNC solution over from discrete
polymatroids representable over Example networks constructed
from discrete polymatroids using the algorithm are provided, which do not admit
any scalar and vector solution, and for which FNC solutions with the message
dimensions being different provide a larger throughput than FNC solutions with
the message dimensions being equal.Comment: 8 pages, 5 figures, 2 tables. arXiv admin note: substantial text
overlap with arXiv:1301.300
Vector Linear Error Correcting Index Codes and Discrete Polymatroids
The connection between index coding and matroid theory have been well studied
in the recent past. El Rouayheb et al. established a connection between multi
linear representation of matroids and wireless index coding. Muralidharan and
Rajan showed that a vector linear solution to an index coding problem exists if
and only if there exists a representable discrete polymatroid satisfying
certain conditions. Recently index coding with erroneous transmission was
considered by Dau et al.. Error correcting index codes in which all receivers
are able to correct a fixed number of errors was studied. In this paper we
consider a more general scenario in which each receiver is able to correct a
desired number of errors, calling such index codes differential error
correcting index codes. We show that vector linear differential error
correcting index code exists if and only if there exists a representable
discrete polymatroid satisfying certain conditionsComment: arXiv admin note: substantial text overlap with arXiv:1501.0506
On the representability of integer polymatroids: Applications in linear code construction
It has been shown that there is a duality between the linear network coding solution and the entropic vectors induced by collection of subspaces in a vector space over a finite field (dubbed linearly constructed entropic vectors). The region of all linearly constructed vectors, coincides with the set of all representable polymatroids. For any integer polymatroid, there is an associated matroid, which uniquely identifies the polymatroid. We conjecture that the representability of the underlying matroid is a sufficient condition for integer polymatroids to be linearly representable. We prove that the conjecture holds for representation over real numbers. Furthermore, we show that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid
On the Fundamental Limits and Symmetric Designs for Distributed Information Systems
Many multi-terminal communication networks, content delivery networks, cache networks, and distributed storage systems can be modeled as a broadcast network. An explicit characterization of the capacity region of the general network coding problem is one of the best known open problems in network information theory. A simple set of bounds that are often used in the literature to show that certain rate tuples are infeasible are based on the graph-theoretic notion of cut. The standard cut-set bounds, however, are known to be loose in general when there are multiple messages to be communicated in the network. This dissertation focuses on broadcast networks, for which the standard cut-set bounds are closely related to union as a specific set operation to combine different simple cuts of the network. A new set of explicit network coding bounds, which combine different simple cuts of the network via a variety of set operations (not just the union), are established via their connections to extremal inequalities for submodular functions. The tightness of these bounds are demonstrated via applications to combination networks.
The tightness of generalized cut-set bounds has been further explored by studying the problem of “latency capacity region” for a broadcast channel. An implicit characterization of this region has been proved by Tian, where a rate splitting based scheme was shown to be optimal. However, the explicit characterization of this region was only available when the number of receivers are less than three. In this dissertation, a precise polyhedral description of this region for a symmetric broadcast channel with complete message set and arbitrary number of users has been established. It has been shown that a set of generalized cut-set bounds, characterizes the entire symmetrical multicast region. The achievability part is proved by showing that every maximum rate vector is feasible by using a successive encoding scheme. The framework for achievability strongly relies on polyhedral combinatorics and it can be useful in network information theory problems when a polyhedral description of a region is needed.
Moreover, it is known that there is a direct relationship between network coding solution and characterization of entropy region. This dissertation, also studies the symmetric structures in network coding problems and their relation with symmetrical projections of entropy region and introduces new aspects of entropy inequalities. First, inequalities relating average joint entropies rather than entropies over individual subsets are studied. Second, the existence of non-Shannon type inequalities under partial symmetry is studied using the concepts of Shannon and non-Shannon groups. Finally, due to the relationship between linear entropic vectors and representability of integer polymatroids, construction of such vector has been discussed. Specifically, It is shown that representability of the particularly constructed matroid is a sufficient condition for integer polymatroids to be linearly representable over real numbers. Furthermore, it has been shown that any real-valued submodular function (such as Shannon entropy) can be approximated (arbitrarily close) by an integer polymatroid