Approximation of graph edit distance in polynomial time enables us to compare large, arbitrarily labeled graphs for structural pattern recognition. In a recent approximation framework, bipartite graph matching (BP) has been proposed to reduce the problem of edit distance to a cubic-time linear sum assignment problem (LSAP) between local substructures. Following the same line of research, first attempts towards quadratic-time approximation have been made recently, including a lower bound based on Hausdorff matching (Hausdorff Edit Distance) and an upper bound based on greedy assignment (Greedy Edit Distance). In this paper, we compare the two approaches and derive a novel upper bound (BP2) which combines advantages of both. In an experimental evaluation on the IAM graph database repository, we demonstrate that the proposed quadratic-time methods perform equally well or, quite surprisingly, in some cases even better than the cubic-time method.