3.2 matchings and factors: algorithms and applications this copyrighted material is taken from...

3.2 Matchings and Factors: Algorithms and Applications

This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course.

These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553.

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3.2.1. Augmenting Path Algorithm Overview

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 (i.e., ’(G)=(G)).

3.2.1 Augmenting path algorithm (for bipartite graphs)

Input An X,Y-bigraph G, a partial matching M, unsaturated U X

Output One of two possibilities:An augmenting path from U to some y YA minimum vertex cover Q

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Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

Maximum Matching/Minimum Cover Algorithm

3.2.1 Augmenting path algorithm (for bipartite graphs)

Input An X,Y-bigraph G, a partial matching M, unsaturated U X

Output One of two possibilities:An augmenting path from U to some y YA minimum vertex cover Q

Maximum Matching/Minimum Cover Algorithm

Input An X,Y-bigraph G, a partial matching M (e.g., M=)Output a maximum matching and a minimum vertex coverRun Algorithm 3.2.1 on M (’(G)–M) times, using the augmenting path output to increase M each time until |M|=’(G) Run Algorithm 3.2.1 once more to produce a minimum vertex cover

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3.2.1. Augmenting Path Algorithm Framework

Algorithm (Augmenting path algorithm)

Input An X,Y-bigraph G, a (partial matching) M, and

the M-unsaturated vertices U XIdea Explore augmenting paths to all possible vertices, marking explored vertices and their predecessors, and tracking reached vertices S X and T YInitialization S=U and T=Iteration [Details next slide]

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3.2.1. Augmenting Path Algorithm Details

Algorithm (Augmenting path algorithm)

Input An X,Y-bigraph G, a (partial matching) M, and

the M-unsaturated vertices U XInitialization S=U and T=Iteration If S is all marked, stop and output M and Q=T (X–S).

Otherwise, explore from an unmarked x S.

For each y N(x) with xy M:

(1) If y is unsaturated, halt and report an augmenting U,y-path

(2) If y is saturated, for the unique edge yw M:(1) Update T T {y}

(2) Update S S {w}

(3) Mark x and iterate

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Augmenting path algorithm example

x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

Run 1 Input: M= S=U=X, T=, x=x1

6

SSS S S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

Run 1: y1 unsaturated. Halt and augment M.

7

T

SSS S S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

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Run 2 Input: M={x1y1} S={x2, x3, x4, x5}, T=, x=x2

SSS S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

9

Run 2: From x2, explore y1 and its matched neighbor

T

SSS S S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

10

Run 2: From x1, explore y2.y2 is unsaturated: halt and augment M.

T

SSS S S

T



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

11

Run 3 Input: M={x1y2, x2y1} S={x3, x4, x5}, T=, x=x3

SS S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

12

Run 3: From x3, explore y3.y3 unsaturated; halt and report augmenting path.

SS S

T



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

13

Run 4 Input: M={x1y2, x2y1, x3y3} S={x4, x5}, T=, x=x4

S S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

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Run 4: From x4, explore y2,y3 and their (distinct) matched neighbors.

SSS

T T

S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

15

Run 4: From x1, and x3, explore to find y1 via non-matching edges. Explore back up to x2 via a matching edge.

SSS

T T

S

T

S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

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Run 4 (continued): x5 is still unexplored. Explore from x5 to find unsaturated vertex y4. Terminate and report an M-augmenting path.

SSS S

T TT

S

T



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

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Run 5 Input: M={x1y2,x2y1,x3y3,x5y4} S={x4}, T=, x=x4

S



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

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Run 5: From x4, explore y2,y3 and their (distinct) matched neighbors.

SSS

T T



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

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Run 5: From x1, and x3, explore to find y1 via non-matching edges. Explore back up to x2 via a matching edge.

SSS S

T TT



x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

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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}.

SSS S

T TT

Q

QQQ


Proof of Repeated Augmenting Path Algorithm

3.2.2 Theorem Repeatedly applying the Augmenting Path Algorithm to a bipartite graph produces a matching and a vertex cover of equal size.

Proof Repeatedly run Algorithm 3.2.1, increasing the matching size by 1 each time, until the first time an augmenting path is not returned. We must argue that the set Q returned by Algorithm 3.2.1 is a vertex cover with size equal to that of the matching M initializing this last run.

We must argue that Q=T (X–S) is a vertex cover. It suffices to prove there are no edges of G between S and Y–T.(1) There is no matching edge of M between S and Y–T:There is certainly no matching edge incident to U. (unsaturated)Vertices of S–U only have matching edges to T.

(2) There is no non-matching edge of E(G)–M between S and Y–T:




Proof (continued) (1) There is no matching edge of M between S and Y–T:There is certainly no matching edge incident to U. (unsaturated)Vertices of S–U only have matching edges to T.

(2) There is no non-matching edge of E(G)–M between S and Y–T: While exploring vertices in S, all non-matching edges are followed so that the other endpoints are placed in T.

(3) Q=T (X-S) has |Q| = |M|:




Proof (continued) (1) There is no matching edge of M between S and Y–T.(2) There is no non-matching edge of E(G)–M between S and Y–T.

(3) Q=T (X–S) has |Q| = |M|:T is involved in exactly T matching edges since these vertices wereexplored through augmenting paths from S.

X–S is saturated; otherwise these vertices would contribute to U.There are no edges from T to X–S because T matches to S–U.Therefore |Q| = |T| + |X–S| = |M|.


Weighted Bipartite Matching

In weighted bipartite matching, we have a simple bigraph with weighted edges and want a matching with maximum possible total edge weight.

Input A simple X,Y-bigraph G and a weight function w:E(G)[0, )Output A matching M with maximum w(M)=e M w(e)

Without loss of generality, we may assume G = Kn,n by adding any missing vertices needed to make |X|=|Y| and assigning edge weight 0 to any missing edges.

Comments• Edge weights of Kn,n are encoded in an nn matrix • Maximum bipartite matching can be solved by maximum weighted

bipartite matching, by assigning edge weight 1 to all edges.• We will define weighted covers, which generalize vertex covers.


Transversals and (Weighted) Covers

0 0 0 0 0

6 4 4 4 3 6

4 1 1 4 3 4

5 1 4 5 3 5

9 5 6 4 7 9

8 5 3 6 8 3

Transversal: A set M of n entries of an nn, 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.

u

v

w(M)=23

w(u,v)=32


v1 v2 v3 v4 v5

u1

u2

u3

u4

u5

3.2.7 Maximum Transversals and Minimum Covers

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.

0 1 2

4 4 4 4

2 1 1 4

3 1 4 5

u

v

w(M)=12

w(u,v)=12


Finding Maximum Transversals and Minimum Covers

We know:

1. We can always find a cover by setting ui =maxj wi,j for all rows i=1,…,n

2. If we find M, (u,v) with w(M)=c(u,v), by Lemma 3.2.7 we are done.

Question But how do we get from Item 1 to Item 2?

Answer Compare c(u,v) versus the weights matrix [wi,j] to

• Make the equality subgraph of edges for which wi,j=ui+vj

• Improve (u,v) using a vertex cover of the equality subgraph

• Iterate until w(M)=c(u,v). Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

The Equality Subgraph

3.2.8 Definition. The equality subgraph Gu,v for a weighted cover (u,v) is the spanning subgraph of Kn,n whose edges are the pairs xiyj such that wi,j=ui+vj. In the cover, the excess for i,j is ui + vj - wi,j.


0 0 0 0 06 4 1 6 2 3

7 5 0 3 7 6

8 2 3 4 5 8

6 3 4 6 3 4

8 4 6 5 8 6

u

x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

vEquality subgraph for (u,v)

Matrix entries with wi,j=ui+vj

The Equality Subgraph and Excess Matrix


0 0 0 0 06 4 1 6 2 3

7 5 0 3 7 6

8 2 3 4 5 8

6 3 4 6 3 4

8 4 6 5 8 6

u

x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

v

Equality subgraph for (u,v)

Matrix entries with wi,j=ui+vj

uv

excess matrix for (u,v)

0 0 0 0 0

6 2 5 0 4 3

7 2 7 4 0 1

8 6 5 4 3 0

6 3 2 0 3 2

8 4 2 3 0 2

3.2.9. Hungarian Algorithm (Maximum Weighted Matching/Minimum Weighted Cover)

Algorithm (Hungarian Algorithm)

Input An n x 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=maxi wi,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 ui ui- for all xi X-R.

Replace vj vj+ for all yj Y-T.

Form the new equality subgraph Gu,v and repeat.

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Hungarian Algorithm Example

0 0 0 0 06 4 1 6 2 3

7 5 0 3 7 6

8 2 3 4 5 8

6 3 4 6 3 4

8 4 6 5 8 6

uv

w(M)=21 w(u,v)=35

0 0 0 0 0

6 2 5 0 4 3

7 2 7 4 0 1

8 6 5 4 3 0

6 3 2 0 3 2

8 4 2 3 0 2

uv excess matrix

T T

R

x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

T T

R

Equality subgraphvertex cover: Q=R T

Add =1 to T, subtract from X–R.



0 0 0 0 06 4 1 6 2 3

7 5 0 3 7 6

8 2 3 4 5 8

6 3 4 6 3 4

8 4 6 5 8 6

uv

w(u,v)=35T T

R



0 0 1 1 05 4 1 6 2 3

6 5 0 3 7 6

8 2 3 4 5 8

5 3 4 6 3 4

7 4 6 5 8 6

uv

w(u,v)=33

The total cover weight must decrease becausenew cover weight – old cover weight = (|T| – |X–R|)

Note: |T| – |X–R| = |T| – (n – |R|) =|T| + |R| – n < n – n = 0 So: new cover weight – old cover weight < 0.


0 0 0 0 06 4 1 6 2 3

7 5 0 3 7 6

8 2 3 4 5 8

6 3 4 6 3 4

8 4 6 5 8 6

uv

w(u,v)=35T T

R



0 0 1 1 05 4 1 6 2 3

6 5 0 3 7 6

8 2 3 4 5 8

5 3 4 6 3 4

7 4 6 5 8 6

uv

w(u,v)=33

When the matrix weights are rational, transform the problem by multiplying through by the least common denominator. Then 1 is guaranteed, and the cover weight w(u,v) decreases by at least 1 in each iteration.See the text for the proof when the matrix has an irrational weight.


0 0 1 1 05 4 1 6 2 3

6 5 0 3 7 6

8 2 3 4 5 8

5 3 4 6 3 4

7 4 6 5 8 6

uv

w(M)=21 w(u,v)=33

0 0 1 1 0

5 1 4 0 4 2

6 1 6 4 0 0

8 6 5 5 4 0

5 2 1 0 3 1

7 3 1 3 0 1

uv excess matrix

T T

x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

T T T

Equality subgraphvertex cover: Q=R T

T

Add =1 to T, subtract from X-R.



0 0 2 2 14 4 1 6 2 3

5 5 0 3 7 6

7 2 3 4 5 8

4 3 4 6 3 4

6 4 6 5 8 6

uv

w(M)=31 w(u,v)=31

0 0 2 2 1

4 0 3 0 4 2

5 0 5 4 0 0

7 5 4 5 4 0

4 1 0 0 3 1

6 2 0 3 0 1

uv excess matrix

T T

x1 x2 x3 x4 x5

y1 y2 y3 y4 y5

T

Matching weight=cover weight.Halt and output M, and (u,v).

equality subgraphContains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553.

3.2 matchings and factors: algorithms and applications this copyrighted material is taken from...

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