Let G be an edge-weighted hypergraph on n vertices, m edges of size O(1), where the edges have real weights in an interval [1, W ]. We show that if we can approximate a maximum weight matching in G within factor α in time T (n,m,W ) then we can find a matching of weight at least (α − ǫ) times the maximum weight of a matching in G in time (ǫ) max 1≤q≤O(ǫ log n ǫ log ǫ−1 ) maxm1+...mq=m ∑q 1 T (n...