Near Approximation of Maximum Weight Matching through Efficient Weight Reduction

Abstract

Let G be an edge-weighted hypergraph on n vertices, m edges of size ≤ 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 α in time T(n,m,W) then we can find a matching of weight at least (α − ε) times the maximum weight of a matching in G in time (ε ^− 1)^O(1)× $\max_{1\le q \le O(\epsilon \frac {\log {\frac n {\epsilon}}} {\log \epsilon^{-1}})} \max_{m_1+...m_q=m}\sum_1^qT(\min\{n,sm_j\},m_{j},(\epsilon^{-1})^{O(\epsilon^{-1})}).$ We obtain our result by an approximate reduction of the original problem to $O(\epsilon \frac {\log {\frac n {\epsilon}}} {\log \epsilon^{-1}})$ subproblems with edge weights bounded by $(\epsilon^{-1})^{O(\epsilon^{-1})}.$ In particular, if we combine our result with the recent (1 − ε)-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 − ε)-approximation algorithm for maximum weight matching in graphs running in time (ε ^− 1)^O(1)(m + n).