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Linear Index Coding via Semidefinite Programming†

Published online by Cambridge University Press:  29 November 2013

EDEN CHLAMTÁČ  and
ISHAY HAVIV
EDEN CHLAMTÁČ
Affiliation:
Department of Computer Science, Ben Gurion University of the Negev, PO box 653, Beer Sheva 84105, Israel (e-mail: chlamtac@cs.bgu.ac.il)
ISHAY HAVIV
Affiliation:
School of Computer Science, The Academic College of Tel Aviv–Yaffo, Tel Aviv 61083, Israel
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Abstract

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, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011).

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 Õ(nf(k)), 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 for graph colouring (J. Assoc. Comput. Mach., 1998) 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 Lovász ϑ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

Keywords

Primary 05C85
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 2 , March 2014 , pp. 223 - 247
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

†

A preliminary version appeared in Proceedings of the 23rd Annual ACM–SIAM Symposium on Discrete Algorithms, 2012, pp. 406–419.

References

[1]Ahlswede, R., Cai, N., Li, S.-Y. R. and Yeung, R. W. (2000) Network information flow. IEEE Trans. Inform. Theory 46 1204–1216.CrossRefGoogle Scholar
[2]Alon, N. (2011) Personal communication.Google Scholar
[3]Alon, N., Lubetzky, E., Stav, U., Weinstein, A. and Hassidim, A. (2008) Broadcasting with side information. In FOCS, pp. 823–832.CrossRefGoogle Scholar
[4]Arora, S., Chlamtac, E. and Charikar, M. (2006) New approximation guarantee for chromatic number. In STOC, pp. 215–224.Google Scholar
[5]Bar-Yossef, Z., Birk, Y., Jayram, T. S. and Kol, T. (2011) Index coding with side information. IEEE Trans. Inform. Theory 57 1479–1494. Preliminary version in FOCS'06.Google Scholar
[6]Birk, Y. and Kol, T. (2006) Coding on demand by an informed source (ISCOD) for efficient broadcast of different supplemental data to caching clients. IEEE Trans. Inform. Theory 52 2825–2830. Preliminary version in INFOCOM'98.Google Scholar
[7]Blum, A. (1994) New approximation algorithms for graph coloring. J. Assoc. Comput. Mach. 41 470–516. Preliminary versions in STOC'89 and FOCS'90.Google Scholar
[8]Blum, A. and Karger, D. R. (1997) An O(n 3/14)-coloring algorithm for 3-colorable graphs. Inform. Process. Lett. 61 49–53.CrossRefGoogle Scholar
[9]Charikar, M., Makarychev, K. and Makarychev, Y. (2006) Near-optimal algorithms for unique games. In STOC, pp. 205–214.Google Scholar
[10]Chlamtac, E. (2007) Approximation algorithms using hierarchies of semidefinite programming relaxations. In FOCS, pp. 691–701.Google Scholar
[11]Chlamtac, E. (2009) Non-local analysis of SDP-based approximation algorithms. Dissertation, Princeton University.Google Scholar
[12]Dinur, I. and Shinkar, I. (2010) On the conditional hardness of coloring a 4-colorable graph with super-constant number of colors. In APPROX-RANDOM, pp. 138–151.Google Scholar
[13]Dinur, I., Mossel, E. and Regev, O. (2009) Conditional hardness for approximate coloring. SIAM J. Comput. 39 843–873. Preliminary version in STOC'06.Google Scholar
[14]El Rouayheb, S., Sprintson, A. and Georghiades, C. (2008) On the relation between the index coding and the network coding problems. In ISIT, IEEE Press, pp. 1823–1827.Google Scholar
[15]Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, third edition, Wiley.Google Scholar
[16]Garey, M. R., Graham, R. L. and Johnson, D. S. (1976) Some NP-complete geometric problems. In STOC, pp. 10–22.Google Scholar
[17]Guruswami, V. and Khanna, S. (2004) On the hardness of 4-coloring a 3-colorable graph. SIAM J. Discrete Math. 18 30–40. Preliminary version in CCC'00.Google Scholar
[18]Haemers, W. (1979) On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 231–232.Google Scholar
[19]Haemers, W. (1981) An upper bound for the Shannon capacity of a graph. In Algebraic Methods in Graph Theory, Vols I, II (Szeged, 1978), Vol. 25 of Colloq. Math. Soc. János Bolyai, North-Holland, pp. 267–272.Google Scholar
[20]Halperin, E., Nathaniel, R. and Zwick, U. (2002) Coloring k-colorable graphs using relatively small palettes. J. Algorithms 45 72–90. Preliminary version in SODA'01.CrossRefGoogle Scholar
[21]Karger, D. R., Motwani, R. and Sudan, M. (1998) Approximate graph coloring by semidefinite programming. J. Assoc. Comput. Mach. 45 246–265. Preliminary version in FOCS'94.Google Scholar
[22]Khanna, S., Linial, N. and Safra, S. (2000) On the hardness of approximating the chromatic number. Combinatorica 20 393–415. Preliminary version in ISTCS'93.Google Scholar
[23]Khot, S. (2001) Improved inapproximability results for maxclique, chromatic number and approximate graph coloring. In FOCS, pp. 600–609.Google Scholar
[24]Khot, S. (2002) On the power of unique 2-prover 1-round games. In STOC, pp. 767–775.CrossRefGoogle Scholar
[25]Knuth, D. E.The sandwich theorem. (1994) Electron. J. Combin. 1.Google Scholar
[26]Langberg, M. and Sprintson, A. (2011) On the hardness of approximating the network coding capacity. IEEE Trans. Inform. Theory 57 1008–1014. Preliminary version in ISIT'08.Google Scholar
[27]Lovász, L. (1979) On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 1–7.Google Scholar
[28]Lubetzky, E. and Stav, U. (2009) Nonlinear index coding outperforming the linear optimum. IEEE Trans. Inform. Theory 55 3544–3551. Preliminary version in FOCS'07.Google Scholar
[29]Peeters, R. (1996) Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica 16 417–431.Google Scholar
[30]Shannon, C. E. (1956) The zero error capacity of a noisy channel. IRE Trans. Inform. Theory IT-2 8–19.Google Scholar
[31]Wigderson, A. (1983) Improving the performance guarantee for approximate graph coloring. J. Assoc. Comput. Mach. 30 729–735. Preliminary version in STOC'82.CrossRefGoogle Scholar
[32]Yeung, R. W. and Zhang, Z. (1999) Distributed source coding for satellite communications. IEEE Trans. Inform. Theory 45 1111–1120.Google Scholar
[33]Zuckerman, D. (2007) Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing 3 103–128. Preliminary version in STOC'06.Google Scholar