3.2.1. Augmenting path algorithm

1. 3.2.1. Augmenting path algorithm Two theorems to recall: Theorem 3.1.10 (Berge).A matching M in a graph G is a maximum matching in G iff G has no M-augmenting path. Theorem 3.1.16 (König,Egerváry) If G is bipartite, then a maximum matching and a minimum vertex cover of G have the same size (’(G)=(G)). Augmenting path algorithm for maximum bipartite matching Input: X,Y-bigraph. Output: M,Q with |M|=|Q| Use modified breadth-first search to find augmenting paths. Initialize with empty matching and iteratively increase by 1. Produce a vertex cover of same size to certify the output.

2. 3.2.1. Augmenting path algorithm Algorithm (Augmenting path algorithm) Input An X,Y-bigraph G, a (partial matching) M, and the set U of M-unsaturated vertices of X Idea Explore augmenting paths to all possible vertices, marking explored vertices and their predecessors, and tracking reached vertices SµX and TµY Initialization S=U and T=; Iteration If S is all marked, stop and output M and Q=T[(X-S). Otherwise, explore from an unmarked x2S. For any edge xy2E(G)-M: (1) mark y and put y in T. (2) For any edge yw2M, mark w and put w in S. Stop if an unsaturated y is found; report an M-augmenting path. Otherwise continue exploring in this fashion.

3. Augmenting path algorithm example S S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 Iteration 1 Input: M=;S=U=X, T=;, x=x1

4. Augmenting path algorithm example S S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T Iteration 1: y1 unsaturated. Halt and augment M.

5. Augmenting path algorithm example S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 Iteration 2 Input: M={x1y1} S={x2, x3, x4, x5}, T=;, x=x2

6. Augmenting path algorithm example S S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T Iteration 2: From x2, explore y1 and its matched neighbors

7. Augmenting path algorithm example S S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T T Iteration 2: From x1, explore y2 and its matched neighbors. y2 is unsaturated: halt and augment M.

8. Augmenting path algorithm example S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 Iteration 3 Input: M={x1y2, x2y1} S={x3, x4, x5}, T=;, x=x3

9. Augmenting path algorithm example S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T Iteration 3: From x3, explore y3 and its matched neighbors. y3 unsaturated, so terminate and report augmenting path.

10. Augmenting path algorithm example S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 Iteration 4 Input: M={x1y2, x2y1, x3y3} S={x4, x5}, T=;, x=x4

11. Augmenting path algorithm example S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T T Iteration 4: From x4, explore y2,y3 and their (distinct) matched neighbors.

12. Augmenting path algorithm example S S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T T T Iteration 4: From x1, and x3, explore to find y1 via non-matching edges. Explore back up to x2 via a matching edge.

13. Augmenting path algorithm example S S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T T T T Iteration 4 (continued): x5 is still unexplored. Explore from x5 to find unsaturated vertex y4. Terminate and report an M-augmenting path.

14. Augmenting path algorithm example S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 Iteration 5 Input: M={x1y2,x2y1,x3y3,x5y4} S={x4}, T=;, x=x4

15. Augmenting path algorithm example S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T T Iteration 5: From x4, explore y2,y3 and their (distinct) matched neighbors.

16. Augmenting path algorithm example S S S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T T T Iteration 5: From x1, and x3, explore to find y1 via non-matching edges. Explore back up to x2 via a matching edge.

17. Augmenting path algorithm example S S Q S S x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 T T T Q Q Q Termination: No vertex of S is unexplored. Output: maximum matching M={x1y2,x2y1,x3y3,x5y4} Minimum vertex cover Q=T[(X-S) ={x5,y1,y2,y3}.

18. Maximum Weighted Transversal Transversal: A set M of n entries of an n£n, matrix, no two in the same column or row. Its weight is the sum of the entries. Cover: A pair of vectors (u,v)=(u1,…,un;v1,…,vn) such that every entry wi,j of the matrix satisfies wi,j·ui+vj. v w(u,v)=32 u w(M)=23

19. 3.2.7. Maximum Transversal = Minimum Cover 3.2.7. Lemma. (Duality of weighted matching and weighted cover problems) For a perfect matching M and a weighted cover (u,v) in a weighted bipartite graph G, c(u,v)¸w(M). Also, c(u,v)=w(M) iff M consists of edges xiyj such that ui+vj=wi,j. In this case, M and (u,v) are optimal. v w(u,v)=12 u w(M)=12

20. 3.2.9. Hungarian Algorithm (Maximum Weighted Matching/Minimum Weighted Cover) 20 Algorithm (Hungarian Algorithm) Input An n£n matrix of nonnegative edge weights of Kn,n Idea Iteratively adjust the cover (u,v) until the equality subgraph Gu,v has a perfect matching Initialization Any cover (u,v), such as ui=maxiwi,j and vj=0 Iteration Find a maximum matching M in Gu,v. If M is a perfect matching, stop and output M and (u,v) as a maximum weight matching and minimum weight cover. Otherwise: Let  = smallest excess in Gu,v of an edge from X-R to Y-T. Replace uiÃui- for all xi2X-R. Replace vjÃvj+ for all yj2Y-T. Form the new equality subgraph Gu,v and repeat.

21. Hungarian Algorithm Example excess matrix v v u u  R T T R w(M)=21 w(u,v)=35 x1 x2 x3 x4 x5 Equality subgraph vertex cover: Q=R[T Add =1 to T, subtract from X-R. y1 y2 y3 y4 y5 T T

22. Hungarian Algorithm Example excess matrix v v u u  T T T w(M)=21 w(u,v)=33 x1 x2 x3 x4 x5 Equality subgraph vertex cover: Q=R[T Add =1 to T, subtract from X-R. y1 y2 y3 y4 y5 T T T

23. Hungarian Algorithm Example excess matrix v v u u T T T w(M)=31 w(u,v)=31 x1 x2 x3 x4 x5 Matching weight=cover weight. Halt and output M, and (u,v). y1 y2 y3 y4 y5 equality subgraph