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 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 et al., FOCS, 2006). 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 $\widetilde{O}(n^{f(k)})$, 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 (J. ACM, 1998) for graph coloring 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 Lovasz theta-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

💡 Research Summary

The paper studies the linear index coding problem, where a sender must broadcast an n‑bit word to n receivers, each of which knows a subset of the bits determined by an undirected side‑information graph G. The goal is to minimize the length of a broadcast message that enables every receiver to recover its own bit. For linear index codes the optimal length equals the min‑rank of G over 𝔽₂, a graph parameter introduced by Haemers and later shown to bound the Shannon capacity of a graph.

The authors present a polynomial‑time algorithm that, given an n‑vertex graph G with min‑rank k (k treated as a constant), constructs a linear index code of length (\widetilde{O}(n^{f(k)})), where the exponent f(k) depends only on k. For the first non‑trivial case k = 3 they obtain f(3) ≈ 0.2574, dramatically improving over the naïve O(√n) bound that follows from the fact that any graph of min‑rank 3 has neighborhoods that are bipartite.

The algorithm is based on a semidefinite programming (SDP) relaxation originally introduced by Karger, Motwani and Sudan for graph coloring. The SDP computes the vector chromatic number χᵥ of the complement graph (\overline{G}). While χᵥ is a relaxation of the ordinary chromatic number, it is not a relaxation of min‑rank, so the authors must relate the SDP optimum to the min‑rank parameter. To this end they define a family of extremal graphs (G_k): for each k, (G_k) maximizes the ordinary chromatic number among all graphs whose complement has min‑rank k. They prove that (G_k) also maximizes the vector chromatic number and the Lovász theta‑function among such graphs.

A key technical contribution is an exact formula for the maximum possible Lovász theta‑value of a graph with min‑rank k: \