On Simple Multiple Access Networks

We investigate a simple multiple access network (SMAN) where k independent sources of unit rates multicast their information to a set of sinks, via n commonly shared relays. All links are assumed to have unit capacity. Given such a SMAN, a coding scheme for the relays is called optimal if each sink can retrieve all information from the sources under at most ⌊n-k+1/2⌋ node/link errors. We study the problem of designing the sparsest SMAN, i.e., the SMAN that has the least number of edges, that supports an optimal coding scheme for the relays. Additionally, the SMAN must satisfy either of the following constraints: 1) Connection Constraint: Each relay can be connected only to a given subset of sources or 2) Balance Constraint: Each relay must be connected to approximately the same number of sources. We provide two polynomial time algorithms to identify the cases where such a SMAN exists together with its optimal coding scheme designed over sufficiently large fields. One algorithm is based on a nontrivial modification of the well-known Gale-Ryser algorithm, whereas the other is based on a novel generalization of the famous Hall´s marriage theorem. We also propose a combinatorial approach to construct optimal coding schemes over small fields and settle the problem for a special case.