Linear index coding via semidefinite programming

Eden Chlamtáč, Ishay Haviv

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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(n f(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 ™-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

Original languageEnglish
Pages (from-to)223-247
Number of pages25
JournalCombinatorics Probability and Computing
Issue number2
StatePublished - 1 Mar 2014


  • 2010 Mathematics subject classification: Primary 05C85

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics
  • Applied Mathematics


Dive into the research topics of 'Linear index coding via semidefinite programming'. Together they form a unique fingerprint.

Cite this