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We consider the fairness in cooperative data exchange (CDE) problem among aset of wireless clients. In this system, each client initially obtains a subsetof the packets. They exchange packets in order to reconstruct the entire packetset. We study the problem of how to find a transmission strategy thatdistributes the communication load most evenly in all strategies that have thesame sum-rate (the total number of transmissions) and achieve universalrecovery (the situation when all clients recover the packet set). We formulatethis problem by a discrete minimization problem and prove its $M$-convexity. Weshow that our results can also be proved by the submodularity of the feasibleregion shown in previous works and are closely related to the resourceallocation problems under submodular constraints. To solve this problem, wepropose to use a steepest descent algorithm (SDA) based on $M$-convexity. Byvarying the number of clients and packets, we compare SDA with a deterministicalgorithm (DA) based on submodularity in terms of convergence performance andcomplexity. The results show that for the problem of finding the fairest andminimum sum-rate strategy for the CDE problem SDA is more efficient than DAwhen the number of clients is up to five.

Fairest Constant Sum-rate Transmission for Cooperative Data Exchange: An M-convex Minimization Approach