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We propose a novel technique for constructing a graph representation of acode through which we establish a significant connection between the servicerate problem and the well-known fractional matching problem. Using thisconnection, we show that the service capacity of a coded storage system equalsthe fractional matching number in the graph representation of the code, andthus is lower bounded and upper bounded by the matching number and the vertexcover number, respectively. This is of great interest because if the graphrepresentation of a code is bipartite, then the derived upper and lower boundsare equal, and we obtain the capacity. Leveraging this result, we characterizethe service capacity of the binary simplex code whose graph representation, aswe show, is bipartite. Moreover, we show that the service rate problem can beviewed as a generalization of the multiset primitive batch codes problem.

A Combinatorial View of the Service Rates of Codes Problem, its Equivalence to Fractional Matching and its Connection with Batch Codes