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On the Index Coding Problem and its Relation to Network Coding and Matroid Theory
"... The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless adhoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X = {x1,..., xk} an ..."
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Cited by 57 (5 self)
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The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless adhoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X = {x1,..., xk} and a set of receivers R. Each receiver ρ = (x,H) in R needs to obtain a message x ∈ X and has prior side information consisting of a subset H of X. The sender uses a noiseless communication channel to broadcast encoding of messages in X to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers. In this paper, we analyze the relation between the index coding problem, the more general network coding problem, and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions.
Broadcasting with side information
 Proc. of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008
"... A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt be the minimum number of bits that has t ..."
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Cited by 47 (5 self)
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A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,..., Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let βt be the minimum number of bits that has to be transmitted when each block is of length t, and let β be the limit β = limt→ ∞ βt/t. In words, β is the average communication cost per bit in each block (for long blocks). Finding the coding rate β, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding. In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on βt, and prove that for any constant C there is an explicit broadcast setting in which β = 2 but β1> C. One of these examples answers a question of [15]. In addition, we provide examples with the following counterintuitive directsum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the
On the hardness of approximating the network coding capacity
 In Proc. IEEE Symp. on Inform. Theory (ISIT
, 2008
"... Abstract—This work addresses the computational complexity of achieving the capacity of a general network coding instance. We focus on the linear capacity, namely the capacity of the given instance when restricted to linear encoding functions. It has been shown [Lehman and Lehman, SODA 2005] that de ..."
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Cited by 32 (8 self)
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Abstract—This work addresses the computational complexity of achieving the capacity of a general network coding instance. We focus on the linear capacity, namely the capacity of the given instance when restricted to linear encoding functions. It has been shown [Lehman and Lehman, SODA 2005] that determining the (scalar) linear capacity of a general network coding instance is NPhard. In this work we initiate the study of approximation in this context. Namely, we show that given an instance to the general network coding problem of linear capacity C, constructing a linear code of rate αC for any universal (i.e., independent of the size of the instance) constant α ≤ 1 is “hard”. Specifically, finding such network codes would solve a long standing open problem in the field of graph coloring. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., a general instance in which the underlying graph is planar). We show that even for planar networks this problem remains NPhard. I.
Index coding: An interference alignment perspective
 in International Symposium on Information Theory
, 2012
"... The index coding problem is studied from an interference alignment perspective providing new results as well as new insights into, and generalizations of, previously known results. An equivalence is established between the capacity of the multiple unicast index coding (where each message is desired ..."
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Cited by 32 (9 self)
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The index coding problem is studied from an interference alignment perspective providing new results as well as new insights into, and generalizations of, previously known results. An equivalence is established between the capacity of the multiple unicast index coding (where each message is desired by exactly one receiver), and groupcast index coding (where a message can be desired by multiple receivers), which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings where this question has previously been answered. Necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints, and generalizations to feasibility of rate 1 L+1 per message when each destination desires at least L messages, are similarly obtained. Finally, capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks. The solutions are based on vector linear coding.
Topological interference management through index coding
, 2013
"... While much recent progress on interference networks has come about under the assumption of abundant channel state information at the transmitters (CSIT), a complementary perspective is sought in this work through the study of interference networks with no CSIT except a coarse knowledge of the topolo ..."
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Cited by 30 (14 self)
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While much recent progress on interference networks has come about under the assumption of abundant channel state information at the transmitters (CSIT), a complementary perspective is sought in this work through the study of interference networks with no CSIT except a coarse knowledge of the topology of the network that only allows a distinction between weak and significant channels and no further knowledge of the channel coefficients ’ realizations. Modeled as a degreesoffreedom (DoF) study of a partially connected interference network with no CSIT, the problem is found to have a counterpart in the capacity analysis of wired networks with arbitrary linear network coding at intermediate nodes, under the assumption that the sources are aware only of the end to end topology of the network. The wireless (wired) network DoF (capacity) region, expressed in dimensionless units as a multiple of the DoF (capacity) of a single point to point channel (link), is found to be bounded above by the capacity of an index coding problem where the antidotes graph is the complement of the interference graph of the original network and the bottleneck link capacity is normalized to unity. The problems are shown to be equivalent under linear solutions over the same field. An interference alignment
On the relation between the Index Coding and the Network Coding problems
 IEEE International Symposium on Information Theory (ISIT 2008
"... Abstract — In this paper we show that the Index Coding problem captures several important properties of the more general Network Coding problem. An instance of the Index Coding problem includes a server that holds a set of information messages X = {x1,..., xk} and a set of receivers R. Each receiver ..."
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Cited by 29 (6 self)
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Abstract — In this paper we show that the Index Coding problem captures several important properties of the more general Network Coding problem. An instance of the Index Coding problem includes a server that holds a set of information messages X = {x1,..., xk} and a set of receivers R. Each receiver has some side information, known to the server, represented by a subset of X and demands another subset of X. The server uses a noiseless communication channel to broadcast encodings of messages in X to satisfy the receivers ’ demands. The goal of the server is to find an encoding scheme that requires the minimum number of transmissions. We show that any instance of the Network Coding problem can be efficiently reduced to an instance of the Index Coding problem. Our reduction shows that several important properties of the Network Coding problem carry over to the Index Coding problem. In particular, we prove that both scalar linear and vector linear codes are insufficient for achieving the minimal number of transmissions. I.
An Equivalence between Network Coding and Index Coding
, 2014
"... We show that the network coding and index coding problems are equivalent. This equivalence holds in the general setting which includes linear and nonlinear codes. Specifically, we present an efficient reduction that maps a network coding instance to an index coding instance while preserving feasibi ..."
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Cited by 20 (3 self)
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We show that the network coding and index coding problems are equivalent. This equivalence holds in the general setting which includes linear and nonlinear codes. Specifically, we present an efficient reduction that maps a network coding instance to an index coding instance while preserving feasibility. Previous connections were restricted to the linear case.
On coding for cooperative data exchange
 in Proc. Information Theory Workshop
, 2009
"... Abstract—We consider the problem of data exchange by a group of closelylocated wireless nodes. In this problem each node holds a set of packets and needs to obtain all the packets held by other nodes. Each of the nodes can broadcast the packets in its possession (or a combination thereof) via a noi ..."
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Cited by 19 (6 self)
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Abstract—We consider the problem of data exchange by a group of closelylocated wireless nodes. In this problem each node holds a set of packets and needs to obtain all the packets held by other nodes. Each of the nodes can broadcast the packets in its possession (or a combination thereof) via a noiseless broadcast channel of capacity one packet per channel use. The goal is to minimize the total number of transmissions needed to satisfy the demands of all the nodes, assuming that they can cooperate with each other and are fully aware of the packet sets available to other nodes. This problem arises in several practical settings, such as peertopeer systems and wireless data broadcast. In this paper, we establish upper and lower bounds on the optimal number of transmissions and present an efficient algorithm with provable performance guarantees. The effectiveness of our algorithms is established through numerical simulations. I.