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...

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Vydáno v:Combinatorics, probability & computing Ročník 23; číslo 2; s. 223 - 247
Hlavní autoři: CHLAMTÁČ, EDEN, HAVIV, ISHAY
Médium: Journal Article
Jazyk:angličtina
Vydáno: Cambridge, UK Cambridge University Press 01.03.2014
Témata:
Algorithms
Coding
Combinatorial analysis
Graph coloring
Graphs
Optimization
Receivers
Upper bounds
Primary 05C85
ISSN:0963-5483, 1469-2163
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Shrnutí: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.
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ISSN:0963-5483
1469-2163
DOI:10.1017/S0963548313000564