# Near Approximation of Maximum Weight Matching through Efficient Weight Reduction

• Andrzej Lingas
• Cui Di
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)

## 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).

## Keywords

Approximation Algorithm Edge Weight Basic Interval Maximum Weight Match Bound Degree Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Berman, P.: A d/2 Approximation for Maximum Weight Independent Set in d-Claw Free Graphs. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 214–219. Springer, Heidelberg (2000)
2. 2.
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill Book Company, Boston (2001)
3. 3.
Chan, Y.H., Lau, L.C.: On Linear and Semidefinite Programming Relaxations for Hypergraph Matching. ProcGoogle Scholar
4. 4.
Drake, D., Hougardy, S.: A simple approximation algorithm for the weighted matching problem. Info. Proc. Lett. 85, 211–213 (2003)
5. 5.
Drake, D., Hougardy, S.: Linear time local improvements for weighted matchings in graphs. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, pp. 107–119. Springer, Heidelberg (2003)
6. 6.
Drake, D., Hougardy, S.: Improved linear time approximation algorithms for weighted matchings. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 14–23. Springer, Heidelberg (2003)Google Scholar
7. 7.
Duan, R., Pettie, S.: Approximating Maximum Weight Matching in Near-linear Time. In: Proc. FOCS (2010)Google Scholar
8. 8.
Edmonds, J., Karp, R.M.: Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems. J. ACM 19(2), 248–264 (1972)
9. 9.
Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 23(2), 596–615 (1987)
10. 10.
Gabow, H.N.: Data structures for weighted matching and nearest common ancestors with linking. In: First Annual ACM-SIAM Symposium on Discrete Algorithms(SODA), pp. 434–443 (1990)Google Scholar
11. 11.
Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM J. Comput. 18(5), 1013–1036 (1989)
12. 12.
Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for general graph-matching problems. J. ACM 38(4), 815–853 (1991)
13. 13.
Hastad, J.: Clique is Hard to Approximate within n 1 − ε. Acta Math. 182(1), 105–142 (1999)
14. 14.
Hanke, Hougardy: 3/4 − ε and 4/5 − ε approximate MWM algorithms running in O(m log n) and O(m log 2 n) time University of Bonn, Research Institute for Discrete Mathematics Report No. 101010Google Scholar
15. 15.
Hochbaum, D.S.: Approximating Covering and Packing Problems: Set Cover, Vertex Cover, Independent Set, and Related Problems in Approximation Algorithms for NP-hard Problems. In: Hochbaum, D.S. (ed.) PWS Publishing Company, Boston (1997)Google Scholar
16. 16.
Kao, M.-Y., Lam, T.-W., Sung, W.-K., Ting, H.-F.: A Decomposition Theorem for Maximum Weight Bipartite Matchings with Applications to Evolutionary Trees. In: Nešetřil, J. (ed.) ESA 1999. LNCS, vol. 1643, pp. 438–449. Springer, Heidelberg (1999)Google Scholar
17. 17.
Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Research Logistics Quarterly 2, 83–97 (1955)
18. 18.
Pettie, S., Sanders, P.: A simple linear time 2/3-ε approximation for maximum weight matching. Information Processing Letters 91, 271–276 (2004)
19. 19.
Preis, R.: Linear time 1/2-approximation algorithm for maximum weighted matching in general graphs. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 259–269. Springer, Heidelberg (1999)
20. 20.
Sankowski, P.: Weighted bipartite matching in matrix multiplication time. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 274–285. Springer, Heidelberg (2006)