Near approximation of maximum weight matching through efficient weight reduction

Let G be an edge-weighted hypergraph on n vertices, m edges of size \le s, 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 alpha in time T(n,m,W) then we can find a matching of weight at least (alpha-epsilon) times the maximum weight of a matching in G in time (epsilon^{{-1}){O(1)}max_{1\le} q \le O(epsilon \frac {log {\frac n {epsilon}}} {log epsilon^{-1}})} max{m1+...mq=m} sum1^{qT(min{n,smj},m{j},(epsilon{-1}){O(epsilon{-1})}).} In particular, if we combine our result with the recent (1-\epsilon)-approximation algorithm for maximum weight matching in graphs due to Duan and Pettie whose time complexity has a poly-logarithmic dependence on W then we obtain a (1-\epsilon)-approximation algorithm for maximum weight matching in graphs running in time (epsilon^{{-1}){O(1)}(m+n).}