Minrank of Embedded Index Coding Problems and its Relation to Connectedness of a Bipartite Graph

TL;DR

This paper introduces a simplified minrank parameter for embedded index coding problems, explores their graphical representations, and links the connectedness of bipartite graphs to transmission efficiency.

Contribution

It presents a reduced-complexity definition of minrank, graphical models for EICP, and establishes a relationship between graph connectedness and transmission count.

Findings

New reduced-complexity minrank definition.

Graphical structures analogous to cycles and cliques identified.

Connectedness of bipartite graph affects number of transmissions.

Abstract

This paper deals with embedded index coding problem (EICP), introduced by A. Porter and M. Wootters, which is a decentralized communication problem among users with side information. An alternate definition of the parameter minrank of an EICP, which has reduced computational complexity compared to the existing definition, is presented. A graphical representation for an EICP is given using directed bipartite graphs, called bipartite problem graph, and the side information alone is represented using an undirected bipartite graph called the side information bipartite graph. Inspired by the well-studied single unicast index coding problem (SUICP), graphical structures, similar to cycles and cliques in the side information graph of an SUICP, are identified in the side information bipartite graph of a single unicast embedded index coding problem (SUEICP). Transmission schemes based on these graphical structures, called tree cover scheme and bi-clique cover scheme are also presented for an SUEICP. Also, a relation between connectedness of the side information bipartite graph and the number of transmissions required in a scalar linear solution of an EICP is established.