Optimal index codes with near-extreme rates

The min-rank of a digraph was shown by Bar-Yossef et al. (2006) to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this work, the graphs and digraphs of near-extreme min-ranks are characterized. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. It is also shown that the decision problem of whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time.

[1]  Syed Ali Jafar,et al.  Index coding: An interference alignment perspective , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[2]  Ishay Haviv,et al.  Linear Index Coding via Semidefinite Programming† , 2011, Combinatorics, Probability and Computing.

[3]  Muriel Médard,et al.  XORs in the Air: Practical Wireless Network Coding , 2006, IEEE/ACM Transactions on Networking.

[4]  Adi Shamir,et al.  A Linear Time Algorithm for Finding Minimum Cutsets in Reducible Graphs , 1979, SIAM J. Comput..

[5]  R. Z. Norman,et al.  Some properties of line digraphs , 1960 .

[6]  Vijaya Ramachandran,et al.  A Minimax Arc Theorem for Reducible Flow Graphs , 1990, SIAM J. Discret. Math..

[7]  M.A.R. Chaudhry,et al.  Efficient algorithms for Index Coding , 2008, IEEE INFOCOM Workshops 2008.

[8]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[9]  Michael Langberg,et al.  On linear index coding for random graphs , 2011, 2012 IEEE International Symposium on Information Theory Proceedings.

[10]  Muriel Médard,et al.  An algebraic approach to network coding , 2003, TNET.

[11]  Michael Langberg,et al.  On the Hardness of Approximating the Network Coding Capacity , 2008, IEEE Transactions on Information Theory.

[12]  Uri Stav,et al.  Non-Linear Index Coding Outperforming the Linear Optimum , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[13]  Robin Thomas,et al.  Packing directed circuits exactly , 2010, Comb..

[14]  Muriel Médard,et al.  Joint Scheduling and Instantaneously Decodable Network Coding , 2009, GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference.

[15]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[16]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[17]  Muriel Médard,et al.  Symbol-level network coding for wireless mesh networks , 2008, SIGCOMM '08.

[18]  Yitzhak Birk,et al.  Informed-source coding-on-demand (ISCOD) over broadcast channels , 1998, Proceedings. IEEE INFOCOM '98, the Conference on Computer Communications. Seventeenth Annual Joint Conference of the IEEE Computer and Communications Societies. Gateway to the 21st Century (Cat. No.98.

[19]  Yeow Meng Chee,et al.  Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures , 2013, Algorithmica.

[20]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[21]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract) , 2010, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[22]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[23]  Rudolf Ahlswede,et al.  Network information flow , 2000, IEEE Trans. Inf. Theory.

[24]  Femke Bekius,et al.  The Shannon Capacity of a Graph , 2011 .

[25]  Michael Langberg,et al.  Index coding with outerplanar side information , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[26]  Maciej M. Syslo,et al.  A Labeling Algorithm to Recognize a Line Digraph and Output its Root Graph , 1982, Inf. Process. Lett..

[27]  Robert W. Deming,et al.  Independence numbers of graphs-an extension of the Koenig-Egervary theorem , 1979, Discret. Math..

[28]  Paul D. Seymour,et al.  Packing circuits in eulerian digraphs , 1996, Comb..

[29]  Noga Alon,et al.  Broadcasting with Side Information , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[30]  Cheng Zhan,et al.  Broadcast Scheduling Based on Network Coding in Time Critical Wireless Networks , 2010, 2010 IEEE International Symposium on Network Coding (NetCod).

[31]  Michael Langberg,et al.  On the complementary Index Coding problem , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[32]  Robert D. Kleinberg,et al.  Index coding via linear programming , 2010, ArXiv.

[33]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[34]  Uri Stav,et al.  Non-Linear Index Coding Outperforming the Linear Optimum , 2007, FOCS.

[35]  René Peeters,et al.  Orthogonal representations over finite fields and the chromatic number of graphs , 1996, Comb..

[36]  Ziv Bar-Yossef,et al.  Index Coding With Side Information , 2006, IEEE Transactions on Information Theory.

[37]  Ziv Bar-Yossef,et al.  Index Coding With Side Information , 2011, IEEE Trans. Inf. Theory.

[38]  Yitzhak Birk,et al.  Coding on demand by an informed source (ISCOD) for efficient broadcast of different supplemental data to caching clients , 2006, IEEE Transactions on Information Theory.

[39]  Mary Lou Soffa,et al.  Feedback vertex sets and cyclically reducible graphs , 1985, JACM.

[40]  Alexander Sprintson,et al.  On the Index Coding Problem and Its Relation to Network Coding and Matroid Theory , 2008, IEEE Transactions on Information Theory.

[41]  Robert E. Tarjan.  Testing flow graph reducibility , 1973, STOC '73.

[42]  C. Lucchesi,et al.  A Minimax Theorem for Directed Graphs , 1978 .