Linear Index Coding via Semidefinite Programming

Abstract

In theindex codingproblem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast ann-bit word tonreceivers (one bit per receiver), where the receivers haveside informationrepresented by a graphG. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. Forlinearindex coding, the minimum possible length is known to be equal to a graph parameter calledminrank(Bar-Yossef, Birk, Jayram and Kol,IEEE Trans. Inform. Theory, 2011).We show a polynomial-time algorithm that, given ann-vertex graphGwith minrankk, finds a linear index code forGof lengthÕ(nf(k)), wheref(k) depends only onk. For example, fork= 3 we obtainf(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 anupper boundon 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 minrankk. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.