76,369 research outputs found
Linear Index Coding via Semidefinite Programming
In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the
goal is to broadcast an n bit word to n receivers (one bit per receiver), where
the receivers have side information represented by a graph G. The objective is
to minimize the length of a codeword sent to all receivers which allows each
receiver to learn its bit. For linear index coding, the minimum possible length
is known to be equal to a graph parameter called minrank (Bar-Yossef et al.,
FOCS, 2006).
We show a polynomial time algorithm that, given an n vertex graph G with
minrank k, finds a linear index code for G of length ,
where f(k) depends only on k. For example, for k=3 we obtain f(3) ~ 0.2574. Our
algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani
and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to
Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a
relaxation of the minimization problem we consider, a crucial component of our
analysis is an upper bound on the objective value of the SDP in terms of the
minrank.
At the heart of our analysis lies a combinatorial result which may be of
independent interest. Namely, we show an exact expression for the maximum
possible value of the Lovasz theta-function of a graph with minrank k. This
yields a tight gap between two classical upper bounds on the Shannon capacity
of a graph.Comment: 24 page
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures
The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon
capacity of a graph. This parameter of a graph has recently gained much more
attention from the research community after the work of Bar-Yossef et al.
(2006). In their paper, it was shown that the min-rank of a graph G
characterizes the optimal scalar linear solution of an instance of the Index
Coding with Side Information (ICSI) problem described by the graph G. It was
shown by Peeters (1996) that computing the min-rank of a general graph is an
NP-hard problem. There are very few known families of graphs whose min-ranks
can be found in polynomial time. In this work, we introduce a new family of
graphs with efficiently computed min-ranks. Specifically, we establish a
polynomial time dynamic programming algorithm to compute the min-ranks of
graphs having simple tree structures. Intuitively, such graphs are obtained by
gluing together, in a tree-like structure, any set of graphs for which the
min-ranks can be determined in polynomial time. A polynomial time algorithm to
recognize such graphs is also proposed.Comment: Accepted by Algorithmica, 30 page
Bilayer Low-Density Parity-Check Codes for Decode-and-Forward in Relay Channels
This paper describes an efficient implementation of binning for the relay
channel using low-density parity-check (LDPC) codes. We devise bilayer LDPC
codes to approach the theoretically promised rate of the decode-and-forward
relaying strategy by incorporating relay-generated information bits in
specially designed bilayer graphical code structures. While conventional LDPC
codes are sensitively tuned to operate efficiently at a certain channel
parameter, the proposed bilayer LDPC codes are capable of working at two
different channel parameters and two different rates: that at the relay and at
the destination. To analyze the performance of bilayer LDPC codes, bilayer
density evolution is devised as an extension of the standard density evolution
algorithm. Based on bilayer density evolution, a design methodology is
developed for the bilayer codes in which the degree distribution is iteratively
improved using linear programming. Further, in order to approach the
theoretical decode-and-forward rate for a wide range of channel parameters,
this paper proposes two different forms bilayer codes, the bilayer-expurgated
and bilayer-lengthened codes. It is demonstrated that a properly designed
bilayer LDPC code can achieve an asymptotic infinite-length threshold within
0.24 dB gap to the Shannon limits of two different channels simultaneously for
a wide range of channel parameters. By practical code construction,
finite-length bilayer codes are shown to be able to approach within a 0.6 dB
gap to the theoretical decode-and-forward rate of the relay channel at a block
length of and a bit-error probability (BER) of . Finally, it is
demonstrated that a generalized version of the proposed bilayer code
construction is applicable to relay networks with multiple relays.Comment: Submitted to IEEE Trans. Info. Theor
Highly Robust Error Correction by Convex Programming
This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x ∈ ℝ^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrupted by arbitrary gross errors, and when in addition, all the entries of the codeword are contaminated by smaller errors (e.g., quantization errors).
We show that if one encodes the information as Ax where A ∈
ℝ^(m x n) (m ≥ n) is a suitable coding matrix, there are two decoding schemes that allow the recovery of the block of n pieces of information x with nearly the same accuracy as if no gross errors occurred upon transmission (or equivalently as if one had an oracle supplying perfect information about the sites and amplitudes of the gross errors). Moreover, both decoding strategies are very concrete and only involve solving simple convex optimization programs, either a linear program or a second-order cone program. We complement our study with numerical simulations showing that the encoder/decoder pair performs remarkably well
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