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A Scaling Algorithm for Maximum Weight Matching in Bipartite Graphs

A Scaling Algorithm for Maximum Weight Matching in Bipartite Graphs. Ran Duan and Hsin-Hao Su Speaker: Hsin-Hao Su. Introduction. Task assignment, marriage matching, etc. Max: . 19$. Introduction. Maximum Weight Matching (MWM). 1000. i. u. 1732. 1000. j. v. 1732. 1000. k.

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A Scaling Algorithm for Maximum Weight Matching in Bipartite Graphs

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  1. A Scaling Algorithm for Maximum Weight Matching in Bipartite Graphs Ran Duan and Hsin-Hao Su Speaker: Hsin-Hao Su

  2. Introduction • Task assignment, marriage matching, etc. Max: 19$

  3. Introduction Maximum Weight Matching (MWM) 1000 i u 1732 1000 j v 1732 1000 k x Total weight: 3464

  4. Introduction Maximum Weight Perfect Matching (MWPM) • Every vertex must be matched 1000 i u 1732 1000 j v 1732 1000 k x Total weight: 3000

  5. Reduction Between MWM and MWPM n: # vertices m: # edges W: the largest edge weight • If MWPM in time • MWM in time • If MWM in time • MWPM in time + 1000 + 1732 1000 + 1000 1732 + 1732 1000 + 1000 1732 1000

  6. Problem History • [Kuhn 1955] Hungarian Algorithm • Based on the results of König and Egreváry • Recent discovery of Jacobi’s work in 19th century

  7. Results for MWM n: # vertices m: # edges W: the largest edge weight 2012 This result

  8. Duality and Optimality • Linear Programming Theory • Dual y(u) for every vertex u • M is optimal if and only if 1000 1732 1000 1732 1000

  9. The Hungarian Algorithm 1000 1414 1000 1414 1000

  10. The Hungarian Algorithm 1000 1414 1000 1414 1000

  11. The Hungarian Algorithm 1000 1414 1000 1414 1000

  12. Gabow and Tarjan’s Scaling Algorithm • -tightness •  error

  13. Gabow and Tarjan’s Scaling Algorithm For to do • -tight to -tight End For scales Modified Hungarian algorithm  error < 1  optimal!

  14. Faster Hungarian Search • Idea: do it simultaneously • iterations • time per iter. 1000 1414 1000 1414 1000

  15. Our Approach • Each scale takes time • Gabow & Tarjan: scales Scale: scales Time:

  16. First Half iterations • Choose • iterations suffice to decrease y(u) to zero • -tightness in 1000 1414 1000 1414 1000

  17. Second Half • Balinsky & Gomory: Primal method • Tighten matched edges

  18. Dilworth’s Lemma • Given a partially ordered set with n elements, there exists a chain or an anti-chain of size at least

  19. Anti-Chain • Fix it in linear time

  20. Chain • Fix it in linear time

  21. Not a partially ordered set • an augmenting cycle to increase the weight of the matching increase by at least 1

  22. Win-Win Analysis • Either • Dilworth’s Lemma is applicable • matched edges will be tighten • There exists an augmenting cycle • increases by at least 1 • Each happens at most times • Time: matched edges

  23. Conclusion • An time algorithm for MWM • An time algorithm for MWPM • Reduction: if MWM in time, then MWPM in time • Open problem: • An time algorithm for MWPM?

  24. Thank you!

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