Download presentation

Presentation is loading. Please wait.

Published byRalph Goodman Modified over 2 years ago

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. 1

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. 2

3
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 Iteration 1 Input: M=; S=U=X, T=;, x=x 1 3 S S S SS

4
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 Iteration 1: y 1 unsaturated. Halt and augment M. 4 T S S S SS

5
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 5 Iteration 2 Input: M={x 1 y 1 } S={x 2, x 3, x 4, x 5 }, T=;, x=x 2 S SSS

6
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 6 Iteration 2: From x 2, explore y 1 and its matched neighbors T S S S SS

7
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 7 Iteration 2: From x 1, explore y 2 and its matched neighbors. y 2 is unsaturated: halt and augment M. T S S S SS T

8
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 8 Iteration 3 Input: M={x 1 y 2, x 2 y 1 } S={x 3, x 4, x 5 }, T=;, x=x 3 S SS

9
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 9 Iteration 3: From x 3, explore y 3 and its matched neighbors. y 3 unsaturated, so terminate and report augmenting path. S S S T

10
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 10 Iteration 4 Input: M={x 1 y 2, x 2 y 1, x 3 y 3 } S={x 4, x 5 }, T=;, x=x 4 SS

11
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 11 Iteration 4: From x 4, explore y 2,y 3 and their (distinct) matched neighbors. S S S T T S

12
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 12 Iteration 4: From x 1, and x 3, explore to find y 1 via non- matching edges. Explore back up to x 2 via a matching edge. S S S T T S T S

13
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 13 Iteration 4 (continued): x 5 is still unexplored. Explore from x 5 to find unsaturated vertex y 4. Terminate and report an M-augmenting path. S S S S T TT S T

14
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 14 Iteration 5 Input: M={x 1 y 2,x 2 y 1,x 3 y 3,x 5 y 4 } S={x 4 }, T=;, x=x 4 S

15
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 15 Iteration 5: From x 4, explore y 2,y 3 and their (distinct) matched neighbors. S S S T T

16
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 16 Iteration 5: From x 1, and x 3, explore to find y 1 via non- matching edges. Explore back up to x 2 via a matching edge. S S S S T TT

17
Augmenting path algorithm example x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 17 Termination: No vertex of S is unexplored. Output: maximum matching M={x 1 y 2,x 2 y 1,x 3 y 3,x 5 y 4 } Minimum vertex cover Q=T[(X-S) ={x 5,y 1,y 2,y 3 }. S S S S T TT Q Q QQ

18
Maximum Weighted Transversal 00000 644436 411434 514535 956479 853683 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)=(u 1,…,u n ;v 1,…,v n ) such that every entry w i,j of the matrix satisfies w i,j · u i +v j. u v w(M)=23 w(u,v)=32

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 x i y j such that u i +v j =w i,j. In this case, M and (u,v) are optimal. 012 4444 2114 3145 u v w(M)=12 w(u,v)=12

20
3.2.9. Hungarian Algorithm (Maximum Weighted Matching/Minimum Weighted Cover) Algorithm (Hungarian Algorithm) Input An n£n matrix of nonnegative edge weights of K n,n Idea Iteratively adjust the cover (u,v) until the equality subgraph G u,v has a perfect matching Initialization Any cover (u,v), such as u i =max i w i,j and v j =0 Iteration Find a maximum matching M in G u,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 G u,v of an edge from X-R to Y-T. Replace u i Ã u i - for all x i 2 X-R. Replace v j Ã v j + for all y j 2 Y-T. Form the new equality subgraph G u,v and repeat. 20

21
Hungarian Algorithm Example 00000 641623 750376 823458 634634 846586 u v w(M)=21w(u,v)=35 00000 625043 727401 865430 632032 842302 u v excess matrix TT R x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 TT R Equality subgraph vertex cover: Q=R[T Add =1 to T, subtract from X-R.

22
Hungarian Algorithm Example 00110 541623 650376 823458 534634 746586 u v w(M)=21w(u,v)=33 00110 514042 616400 865540 521031 731301 u v excess matrix TT x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 TT T Equality subgraph vertex cover: Q=R[T T Add =1 to T, subtract from X-R.

23
Hungarian Algorithm Example 00221 441623 550376 723458 434634 646586 u v w(M)=31w(u,v)=31 00221 403042 505400 754540 410031 620301 u v excess matrix TT x1x1 x2x2 x3x3 x4x4 x5x5 y1y1 y2y2 y3y3 y4y4 y5y5 T Matching weight=cover weight. Halt and output M, and (u,v). equality subgraph

Similar presentations

OK

Maximum Bipartite Matching In a graph G, if no M-augmenting path exists, then M is a maximum matching in G. Idea: Iteratively seek augmenting paths to.

Maximum Bipartite Matching In a graph G, if no M-augmenting path exists, then M is a maximum matching in G. Idea: Iteratively seek augmenting paths to.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Seminar report and ppt on google glasses Ppt on bluetooth based smart sensor networks machine Ppt on interest rate derivatives Ppt on art and craft movements Ppt online form 2015 last date Ppt on hard gelatin capsule shell Ppt on quality education fund Ppt online viewer php file Ppt on network switching system Ppt on project proposal writing