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Graph-theoretical approaches to biological network analysis have proven to be effective for small networks but are computationally infeasible for comprehensive genome-scale systems-level elucidation of these networks. The difficulty lies in the NP-hard nature of many global systems biology problems that, in practice, translates to exponential (or worse) run times for finding exact optimal solutions. Moreover, these problems, especially those of an enumerative flavor, are often memory-intensive and must share very large sets of data effectively across many processors. For example, the enumeration of maximal cliques - a core component in gene expression networks analysis, cis regulatory motif finding, and the study of quantitative trait loci for high-throughput molecular phenotypes can result in as many as 3^n/3 maximal cliques for a graph with n vertices. Memory requirements to store those cliques reach terabyte scales even on modest-sized genomes. Emerging hardware architectures with ultra-large globally addressable memory such as the SGI Altix and Cray X1 seem to be well suited for addressing these types of data-intensive problems in systems biology. This paper presents a novel framework that provides exact, parallel and scalable solutions to various graph-theoretical approaches to genome-scale elucidation of biological networks. This framework takes advantage of these large-memory architectures by creating globally addressable bitmap memory indices with potentially high compression rates, fast bitwise-logical operations, and reduced search space. Augmented with recent theoretical advancements based on fixed-parameter tractability, this framework produces computationally feasible performance for genome-scale combinatorial problems of systems biology.