Complexity of the pipeline computation of a family of inner products

We are interested in the minimum time T(S) necessary for computing a family S = { < S_i, S_j >: ? S_i, S_j?R^p, (i, j) ?E } of inner products of order p, on a systolic array of order p × 2. We first prove that the determination of T(S) is equivalent to the partition problem and is thus NP-complete. Then we show that the designing of an algorithm which runs in time T(S) + 1 is equivalent to the problem of finding an undirected bipartite eulerian multigraph with the smallest number of edges, which contains a given undirected bipartite graph, and can therefore be solved in polynomial time.