Weighted Bipartite Matching

1. Weighted Bipartite Matching Lecture 4: Jan 18

2. Weighted Bipartite Matching Given a weighted bipartite graph, find a matching with maximum total weight. B A Not necessarily a maximum size matching.

3. Today’s Plan • Three algorithms • negative cycle algorithm • primal dual algorithm • augmenting path algorithm • Applications

4. First Algorithm Find maximum weight perfect matching How to know if a given matching M is optimal? Idea: Imagine there is a larger matching M* and consider the union of M and M*

5. First Algorithm • Orient the edges (edges in M go up, others go down) • edges in M having positive weights, otherwise negative weights Then M is maximum if and only if there is no negative cycles

6. First Algorithm Key: M is maximum  no negative cycle How to find efficiently?

7. Complexity • At most nW iterations • A negative cycle in time by Floyd Warshall • Total running time • Can choose minimum mean cycle to avoid W

8. Augmenting Path Algorithm • Orient the edges (edges in M go up, others go down) • edges in M having positive weights, otherwise negative weights Find a shortest path M-augmenting path at each step

9. Augmenting Path Algorithm Theorem: each matching is an optimal k-matching. Let the current matching be M Let the shortest M-augmenting path be P Let N be a matching of size |M|+1

10. Complexity • At most n iterations • A shortest path in time by Bellman Ford • Total running time • Can be speeded up by using Dijkstra O(m + nlogn) • Kuhn, Hungarian method

11. Primal Dual Algorithm What is an upper bound of maximum weighted matching? What is a generalization of minimum vertex cover? weighted vertex cover: Minimum weighted vertex cover >= Maximum weighted matching

12. Primal Dual Algorithm Minimum weighted vertex cover >= Maximum weighted matching • Primal Dual • Maintain a weighted matching M • Maintain a weighted vertex cover y • Either increase M or decrease y until they are equal.

13. Primal Dual Algorithm Minimum weighted vertex cover >= Maximum weighted matching • Consider the subgraph formed by tight edges (the equality subgraph). • If there is a tight perfect matching, done. • Otherwise, there is a vertex cover. • Decrease weighted cover to create more tight edges, and repeat.

14. Complexity • Augment at most n times, each O(m) • Decrease weighted cover at most m times, each O(m) • Total running time • Egrevary, Hungarian method

15. Quick Summary • First algorithm, negative cycle, useful idea to consider symmetric difference • Augmenting path algorithm, useful algorithmic technique • Primal dual algorithm, a very general framework • Why primal dual? • How to come up with weighted vertex cover? • Reduction from weighted case to unweighted case

16. Faster Algorithms

17. Bonus Question 2 (40%) Find a maximum weighted k-matching which can be extended to a perfect matching. Relation to red-blue matching?

18. Application of Bipartite Matching Isaac Darek Tom Jerry Marking Tutorials Solutions Newsgroup Job Assignment Problem: Each person is willing to do a subset of jobs. Can you find an assignment so that all jobs are taken care of? • Ad-auction, Google, online matching • Fingerprint matching

19. Application of Bipartite Matching With Hall’s theorem, now you can determine exactly when a partial chessboard can be filled with dominos.

20. Application of Bipartite Matching • Latin Square: a nxn square, the goal is to fill the square • with numbers from 1 to n so that: • Each row contains every number from 1 to n. • Each column contains every number from 1 to n.

21. Application of Bipartite Matching Now suppose you are given a partial Latin Square. Can you always extend it to a Latin Square? With Hall’s theorem, you can prove that the answer is yes.