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We consider a memoryless Gaussian interference channel (GIC) where K single-antenna users communicate with their respective receivers using Gaussian codebooks. Each receiver employs a successive group decoder with a specified complexity constraint, to decode its designated user. It is aware of the coding schemes employed by all other users and may choose to decode some or all of them only if it deems that doing so will aid the decoding of its desired user. For a GIC with predetermined rates for all transmitters, we obtain the minimum outage probability decoding strategy at each receiver which satisfies the imposed complexity constraint and reveals the optimal subset of interferers that must be decoded along with the desired user. We then consider the rate allocation problem over the GIC under successive group decoding and design a sequential rate allocation algorithm which yields a Pareto-optimal rate allocation, and two parallel rate allocation algorithms which yield the symmetric fair rate allocation and the max-min fair rate allocation, respectively. Remarkably, even though the proposed decoding and rate allocation algorithms use ldquogreedyrdquo or myopic subroutines, they achieve globally optimal solutions. Finally, we also propose rate allocation algorithms for a cognitive radio system.