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We consider the scheduling of biprocessor jobs under sum objective (BPSMS). Given a collection of unit-length jobs where each job requires the use of two processors, find a schedule such that no two jobs involving the same processor run concurrently. The objective is to minimize the sum of the completion times of the jobs. Equivalently, we would like to find a sum edge coloring of the given multigraphs, i.e. a partition of its edge set into matchings M1, ..., Mt minimizing Σi=1 t i|Mi|. This problem is APX-hard even in the case of bipartite graphs [M04]. This special case is closely related to the classic open shop scheduling problem. We give a 1.829-approximation algorithm for BPSMS that combines a configuration LP with greedy methods improving the previously best known ratio of 2 [BBH+98]. The algorithm uses the fractions derived from the configuration LP and a non-standard randomized rounding. We also give a purely combinatorial and practical algorithm for the case of simple graphs, with a 1.8861-approximation ratio.