Near Approximation of Maximum Weight Matching through Efficient Weight Reduction

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 - e) times the maximum weight of a matching in G in time (e-1)O(1)× max1≤q≤O(e log n/e/log e-1) maxm1+...mq=m Σ1qT(min{n, smj},mj, (e-1)O(e-1)). We obtain our result by an approximate reduction of the original problem to O(e logn/e/log e-1) subproblems with edge weights bounded by (e-1)O(e-1)). In particular, if we combine our result with the recent (1 - e)-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 - e)-approximation algorithm for maximum weight matching in graphs running in time (e-1)O(1)(m+n).

[1]  Robert E. Tarjan,et al.  Faster Scaling Algorithms for Network Problems , 1989, SIAM J. Comput..

[2]  Piotr Sankowski,et al.  Weighted Bipartite Matching in Matrix Multiplication Time , 2006, ICALP.

[3]  Robert E. Tarjan,et al.  Faster scaling algorithms for general graph matching problems , 1991, JACM.

[4]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[5]  Yuk Hei Chan,et al.  On linear and semidefinite programming relaxations for hypergraph matching , 2010, Mathematical Programming.

[6]  Eli Upfal,et al.  Constructing a perfect matching is in random NC , 1985, STOC '85.

[7]  Piotr Berman,et al.  A d/2 Approximation for Maximum Weight Independent Set in d-Claw Free Graphs , 2000, Nord. J. Comput..

[8]  P. Sanders,et al.  A simpler linear time 2 / 3 − ε approximation for maximum weight matching , 2004 .

[9]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[10]  Stefan Hougardy,et al.  A simple approximation algorithm for the weighted matching problem , 2003, Inf. Process. Lett..

[11]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[12]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[13]  Stefan Hougardy,et al.  Improved Linear Time Approximation Algorithms for Weighted Matchings , 2003, RANDOM-APPROX.

[14]  D. Hochbaum.  Approximating covering and packing problems: set cover, vertex cover, independent set, and related problems , 1996 .

[15]  Ran Duan,et al.  Approximating Maximum Weight Matching in Near-Linear Time , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[16]  Ming-Yang Kao,et al.  A Decomposition Theorem for Maximum Weight Bipartite Matchings with Applications to Evolutionary Trees , 1999, ESA.

[17]  Peter Sanders,et al.  A simpler linear time 2/3-epsilon approximation for maximum weight matching , 2004, Inf. Process. Lett..

[18]  Robert Preis,et al.  Linear Time 1/2-Approximation Algorithm for Maximum Weighted Matching in General Graphs , 1999, STACS.

[19]  Stefan Hougardy,et al.  Linear Time Local Improvements for Weighted Matchings in Graphs , 2003, WEA.

[20]  Harold W. Kuhn,et al.  The Hungarian method for the assignment problem , 1955, 50 Years of Integer Programming.

[21]  J. Håstad.  Clique is hard to approximate within n 1-C , 1996 .

[22]  Harold N. Gabow,et al.  Data structures for weighted matching and nearest common ancestors with linking , 1990, SODA '90.