In the search for efficient solution techniques for the large sparse linear systems that arise in engineering practice, multilevel techniques have demonstrated optimal scaling for a wide class of matrices. Highly optimized algorithms may be developed when detailed knowledge of the origins of the matrix problem are considered; however, robust algebraic approaches allow for fast solution without extensive problem-specific tuning. In this talk, we present recent work on algebraic approaches to finding the fine/coarse partitions necessary within these algorithms. Motivated by theoretical analysis and practical concerns, we develop greedy partitioning algorithms based on diagonal-dominance criteria. Using this partitioning within the algebraic recursive multilevel solver (ARMS) framework, we explore the effects of reordering the fine-scale block using standard fill-reducing techniques.