### Abstract

Index coding has received considerable attention recently motivated in part by applications such as fast video-on-demand and efficient communication in wireless networks and in part by its connection to network coding. Optimal encoding schemes and efficient heuristics were studied in various settings, while also leading to new results for network coding such as improved gaps between linear and non-linear capacity as well as hardness of approximation. The problem of broadcasting with side information, a generalization of the index coding problem, begins with a sender and sets of users and messages. Each user possesses a subset of the messages and desires an additional message from the set. The sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message. The fundamental parameter of interest is the broadcast rate, β the average communication cost for sufficiently long broadcasts. Though there have been many new nontrivial bounds on β by Bar-Yossef (2006), Lubetzky and Stav (2007), Alon (2008), and Blasiak (2011) there was no known polynomial-time algorithm for approximating β within a nontrivial factor, and the exact value of β remained unknown for all nontrivial instances. Using the information theoretic linear program introduced in Blasiak (2011), we give a polynomial-time algorithm for recognizing instances with β = 2 and pinpoint β precisely for various classes of graphs (e.g., various Cayley graphs of cyclic groups). Further, extending ideas from Ramsey theory, we give a polynomial-time algorithm with a nontrivial approximation ratio for computing β. Finally, we provide insight into the quality of previous bounds by giving constructions showing separations between β and the respective bounds. In particular, we construct graphs where β is uniformly bounded while its upper bound derived from the naïve encoding scheme is polynomially worse.

Original language | English (US) |
---|---|

Article number | 6578154 |

Pages (from-to) | 5811-5823 |

Number of pages | 13 |

Journal | IEEE Transactions on Information Theory |

Volume | 59 |

Issue number | 9 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

- Approximation algorithms
- broadcasting
- computer science
- graph theory
- linear code
- network coding
- network theory

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Cite this

*IEEE Transactions on Information Theory*,

*59*(9), 5811-5823. [6578154]. https://doi.org/10.1109/TIT.2013.2264472

**Broadcasting with side information : Bounding and approximating the broadcast rate.** / Blasiak, Anna; Kleinberg, Robert; Lubetzky, Eyal.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 59, no. 9, 6578154, pp. 5811-5823. https://doi.org/10.1109/TIT.2013.2264472

}

TY - JOUR

T1 - Broadcasting with side information

T2 - Bounding and approximating the broadcast rate

AU - Blasiak, Anna

AU - Kleinberg, Robert

AU - Lubetzky, Eyal

PY - 2013

Y1 - 2013

N2 - Index coding has received considerable attention recently motivated in part by applications such as fast video-on-demand and efficient communication in wireless networks and in part by its connection to network coding. Optimal encoding schemes and efficient heuristics were studied in various settings, while also leading to new results for network coding such as improved gaps between linear and non-linear capacity as well as hardness of approximation. The problem of broadcasting with side information, a generalization of the index coding problem, begins with a sender and sets of users and messages. Each user possesses a subset of the messages and desires an additional message from the set. The sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message. The fundamental parameter of interest is the broadcast rate, β the average communication cost for sufficiently long broadcasts. Though there have been many new nontrivial bounds on β by Bar-Yossef (2006), Lubetzky and Stav (2007), Alon (2008), and Blasiak (2011) there was no known polynomial-time algorithm for approximating β within a nontrivial factor, and the exact value of β remained unknown for all nontrivial instances. Using the information theoretic linear program introduced in Blasiak (2011), we give a polynomial-time algorithm for recognizing instances with β = 2 and pinpoint β precisely for various classes of graphs (e.g., various Cayley graphs of cyclic groups). Further, extending ideas from Ramsey theory, we give a polynomial-time algorithm with a nontrivial approximation ratio for computing β. Finally, we provide insight into the quality of previous bounds by giving constructions showing separations between β and the respective bounds. In particular, we construct graphs where β is uniformly bounded while its upper bound derived from the naïve encoding scheme is polynomially worse.

AB - Index coding has received considerable attention recently motivated in part by applications such as fast video-on-demand and efficient communication in wireless networks and in part by its connection to network coding. Optimal encoding schemes and efficient heuristics were studied in various settings, while also leading to new results for network coding such as improved gaps between linear and non-linear capacity as well as hardness of approximation. The problem of broadcasting with side information, a generalization of the index coding problem, begins with a sender and sets of users and messages. Each user possesses a subset of the messages and desires an additional message from the set. The sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message. The fundamental parameter of interest is the broadcast rate, β the average communication cost for sufficiently long broadcasts. Though there have been many new nontrivial bounds on β by Bar-Yossef (2006), Lubetzky and Stav (2007), Alon (2008), and Blasiak (2011) there was no known polynomial-time algorithm for approximating β within a nontrivial factor, and the exact value of β remained unknown for all nontrivial instances. Using the information theoretic linear program introduced in Blasiak (2011), we give a polynomial-time algorithm for recognizing instances with β = 2 and pinpoint β precisely for various classes of graphs (e.g., various Cayley graphs of cyclic groups). Further, extending ideas from Ramsey theory, we give a polynomial-time algorithm with a nontrivial approximation ratio for computing β. Finally, we provide insight into the quality of previous bounds by giving constructions showing separations between β and the respective bounds. In particular, we construct graphs where β is uniformly bounded while its upper bound derived from the naïve encoding scheme is polynomially worse.

KW - Approximation algorithms

KW - broadcasting

KW - computer science

KW - graph theory

KW - linear code

KW - network coding

KW - network theory

UR - http://www.scopus.com/inward/record.url?scp=84882737834&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84882737834&partnerID=8YFLogxK

U2 - 10.1109/TIT.2013.2264472

DO - 10.1109/TIT.2013.2264472

M3 - Article

VL - 59

SP - 5811

EP - 5823

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 9

M1 - 6578154

ER -