Linear index coding via graph homomorphism

Abstract

In (1), (2) it is shown that the minimum broadcast rate of a linear index code over a finite field Fq is equal to an algebraic invariant of the underlying digraph, called minrankq. In (3), it is proved that for F2 and any positive integer k, minrankq(G) k if and only if there exists a homomorphism from the complement of the G to the complement of a particular undirected family called graph familyfGkg. As observed in (2), by combining these two results one can relate the linear index coding problem of undirected graphs to the homomorphism problem. In (4), a direct connection between linear index coding problem and homomorphism problem is introduced. In contrast to the former approach, the direct connection holds for digraphs as well and applies to any field size. More precisely, in (4), a familyfH q k g has been introduced and shown that whether or not the scalar linear index of a digraph G is less than or equal to k is equivalent to the existence of a homomorphism from the complement of G to the complement of H q . In this paper, we first study the structure of the digraphs H q k defined in (4). Analogous to the result of (2) about undirected graphs, we prove that H q k 's are vertex transitive digraphs. Using this, and by applying a lemma of Hell and Nesetril (5), we derive a class of necessary conditions for digraphs G to satisfy lindq(G) k. Particularly, we obtain new lower bounds on lindq(G). Our next result is about the computational complexity of scalar linear index of a digraph. It is known that deciding whether the scalar linear index of an undirected is equal to k or not is NP-complete for k 3 and is polynomially decidable for k = 1;2 (3). For digraphs, it is shown in (6) that for the binary alphabet, the decision problem for k = 2 is NP-complete. We use homomorphism framework to extend this result to arbitrary alphabet. Index Terms—Index coding, linear index coding, homo- morphism, minrank of a graph, computational complexity of the minrank. q k) in whichjGj and !(G) stand for the number of vertices and the clique number of G, respectively. We find a lower