Greedy Algorithm And Edmonds Matroid Intersection Algorithmwilhelm/greedy-vortrag.pdf · Greedy Algorithm Transversal Matroids Edmonds’ Intersection Algorithm References Content

Greedy Algorithm Transversal Matroids Edmonds Intersection Algorithm References

Greedy AlgorithmAnd

Edmonds Matroid Intersection Algorithm

Paul [email protected]

Institut fur MathematikHumboldt-Universitat zu Berlin

July 17, 2010

Paul Wilhelm (Math HU Berlin) Greedy Algorithm July 17, 2010 1 / 32

Content I

1 Greedy AlgorithmPrerequisitesAlgorithmFeasibility In Matroids

Proof Greedy Matroid

Proof Matroid Greedy

2 Transversal MatroidsJob Scheduling

Job Scheduling Problem

Mathematical Describtion

Transversal MatroidsDefinition Transversal Matroids

Proof

Greedy Algorithm In Transversal Matroids

3 Edmonds Intersection AlgorithmPaul Wilhelm (Math HU Berlin) Greedy Algorithm July 17, 2010 2 / 32

Content II

PrerequisitesExample Intersection Of Two Matriods

AlgorithmCounterexample Intersection Of Three Matriods

Complexity Of Edmonds Algorithm

OutlookCorrectnes of the Algorithm

Prerequisites

Remind

Independence System / Matroid (E ,I)

I (nonempty)

Y I, X Y X I (hereditary)

X ,Y I, |X | < |Y | y Y \ X : X {y} I (augmentation)

Prerequisites

Remind

I (nonempty)

Prerequisites

Remind

I (nonempty)

Prerequisites

Min/Max - Problem

(due to (Korte and Vygen, 2007, p. 306))

Maximization Problem

Find X I s.t. the weight w(X ) =

eX

w(e) is maximum

Minimization Problem

Find a basis B s.t. the weight w(X ) =

eX

w(e) is minimum

Prerequisites

Min/Max - Problem

eX

w(e) is maximum

eX

w(e) is minimum

Prerequisites

Min/Max - Problem

eX

w(e) is maximum

eX

w(e) is minimum

Prerequisites

Examples

Many combinatorial optimization problems can be formulated asmin/max-problem (more in (Korte and Vygen, 2007, p. 306)):

maximum weight stable set

travelling salesman

shortest path

knapsack

minimum spanning tree

maximum weight forest

steiner tree

maximum weight branching

maximum weight matching

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Examples

travelling salesman

shortest path

knapsack

steiner tree

job scheduling

Prerequisites

Oracle

The independence system (E ,I) is given by an oracle:

Independence Oracle

An oracle which decides whether F E is in I or not.

F E O(F ) :=

{

1 ,if F I0 ,else

Basis-Superset Oracle

An oracle which decides whether B E is an basis for I or not.

B E O(B) :=

{

1 ,if B I and x E : B {x} I0 ,else

Prerequisites

Oracle

Independence Oracle

F E O(F ) :=

{

1 ,if F I0 ,else

B E O(B) :=

{

Prerequisites

Oracle

Independence Oracle

F E O(F ) :=

{

1 ,if F I0 ,else

B E O(B) :=

{

Prerequisites

Oracle

Independence Oracle

F E O(F ) :=

{

1 ,if F I0 ,else

B E O(B) :=

{

Prerequisites

Oracle

Independence Oracle

F E O(F ) :=

{

1 ,if F I0 ,else

B E O(B) :=

{

Prerequisites

Oracle

Independence Oracle

F E O(F ) :=

{

1 ,if F I0 ,else

B E O(B) :=

{

Prerequisites

Oracle

Independence Oracle

F E O(F ) :=

{

1 ,if F I0 ,else

B E O(B) :=

{

Algorithm

Best-In-Greedy

Sort E = {e1, ..., en} s.t. w(e1) ... w(en)

Set F := I

For i:=1 to n do:

If F {ei} I then set F := F {ei}

Output: F I

Sort the elements in E (O(n log(n)))

Have to ask oracle in each step! Complexity of algorithm dependsmainly on complexity of oracle (O(O)).

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Algorithm

Best-In-Greedy

Set F := I

For i:=1 to n do:

Output: F I

Feasibility In Matroids

Theorem Feasibility

Asume the greedy algorithm work correctly in the independence system(E ,I).

Theorem

An independence system is a matroid iff the best-in-greedy algorithm findsfor all weight functions an optimum solution for the maximization problem.

If not a matroid:

find only locally optimum

cardinallity dont have to be maximum, just maximal (i.e. there existsno superset)

Theorem Feasibility

Theorem

If not a matroid:

Theorem Feasibility

Theorem

If not a matroid:

Theorem Feasibility

Theorem

If not a matroid:

Theorem Feasibility

Theorem

If not a matroid:

Proof Greedy Matroid By Contradiction

Assumption: (E ,I) Is No Matroid:

i.e. X ,Y I with |X | < |Y | s.t. e Y \ X : X {e} / I

Define Weight Function:

w(e) :=

1 + , if e X1 , if e Y \ X0 , if e E \ {X Y }

greedy outputs F with w(F ) = |X | (1 + ) + 0

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

to w(F ) maximum for all weight functions.

w(e) :=

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

w(e) :=

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

w(e) :=

1 + , if e X choose first |X | steps1 , if e Y \ X0 , if e E \ {X Y }

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

w(e) :=

1 + , if e X choose first |X | steps1 , if e Y \ X cant choose0 , if e E \ {X Y }

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

w(e) :=

1 + , if e X choose first |X | steps1 , if e Y \ X cant choose0 , if e E \ {X Y } dont change the weight

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

w(e) :=

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

w(e) :=

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

w(e) :=

w(F ) = |X |(1 + ) < w(Y ) = |Y | for < |Y ||X | 1

let w be an abitrary weight function

let F = {f1, ..., fr} the output of the greedy algorithm

w.l.o.g. w(f1) ... w(fr ) (if not change numbering)

first prove that F has maximum cardinality

then prove by contradiction that w(F ) is maximum

Proof Maximum Cardinality Of F By Contradiction

Assume G I s.t. |F | < |G |Augmentation property g G \ F s.t. F {g} t s.t. {f1, ..., ft , g , ft+1, ..., fs} = F {g} I withw(ft) w(g) w(ft+1) {f1, ..., ft} {f1, ..., ft , g} Ig should been choosen in step t + 1 of greedy algorithm

Proof w(F ) is maximum in (E ,I) by contradiction

Assume G = {g1, ..., gr} I s.t. w(G ) > w(F ) and w(gi ) w(gi+1)

giG

w(gi ) >

fiF

w(fi )

k : w(gk) > w(fk) (because |G | |F |)take X := {f1, ..., fk1} (= if k = 1) and Y := {g1, ..., gk}|X | < |Y | (augmentation property) gt Y \ X with t k s.t.{f1, ..., fk1, gt} = X {gt} IBecause w(gt) w(gk) > w(fk) gt should been choosen before step kof the greedy algorithm

to correctness of greedy algorithm

giG

w(gi ) >

fiF

w(fi )

giG

w(gi ) >

fiF

w(fi )

giG

w(gi ) >

fiF

w(fi )

giG

w(gi ) >

fiF

w(fi )

giG

w(gi ) >

fiF

w(fi )

giG

w(gi ) >

fiF

w(fi )

giG

w(gi ) >

fiF

w(fi )

giG

w(gi ) >

fiF

w(fi )

Job Scheduling

Example Job Scheduling

Set of one worker jobs

Arranged in order of importance

Pool of workers

Each is qualified to do a subset of jobs

Jobs have to be done simultaneously, so every worker can do at leastone job

if not all jobs can be done, try optimal relative to the priority

Job Scheduling

Pool of workers

Job Scheduling

Pool of workers

Job Scheduling

Pool of workers

Job Scheduling

Pool of workers

Job Scheduling

Pool of workers

Job Scheduling

Pool of workers

Job Scheduling

Figure 1.19 from (Oxley, 2006, p. 47)

J-set of jobs in order of priority

W-set of workers

Wi is the set of jobs worker i can do

W := {Wi |i I , Wi J} (family ofsubsets)

Vertex set J W and the edge set{jWi |j J,Wi W} define the graph [W]

Job Scheduling

J

j1

j2

j3

j4

j5

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W-set of workers

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

T J partial transversal : injectivemap : T I s.t. t W(t) t T

i.e. matching for T in [W]

transversal iff |T | = |W|

The cardinality of the biggest (partial)transversal is the maximum number of jobsthat can be done simultaneously in theexample.

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Job Scheduling

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

W := {Wi |i I , Wi J}

Transversal Matroids

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

(J,I) Transversal Matroid ofW := {Wi |i I , Wi J} iff I is the set ofpartial transversals of W

Proof:

I1 : because is transversal of thesubfamily W

I2 : I (partial) transversal of W and I Ithen I is a partial transversal of W

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

Proof:

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

Proof:

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

Proof:

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

Proof:

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

Proof:

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

Proof:

Definition

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Transversal Matroid

Proof:

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

I1, I2 partial transversals of W with |I1| < |I2|

matchings in [W]:

M1 match I1 into W1 and M2 match I2 into W2

Searching matching for I1 {e} (e I2 \ I1)

|I1| = |M1| < |I2| = |M2| more blue

M1 \ M2, M2 \ M1 induce subgraph M of [W]

M consist of two matchings in a bipartite graph (v) {1, 2} vertices v in M and no twoedges of same color meet in a vertex

Every connected component of M is even path,odd path or even cycle (no odd cycle or tree)

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3 (|I1| < |I2| : e I2 \ I1 I1 {e} I):

matchings in [W]:

|I1| = |M1| < |I2| = |M2| more blue

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

Even path and even cycle: equal number of colors

more blue odd path P with blue first andlast edge

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

more blue odd path P with blue first andlast edgechange example

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

If v1 J vi J (for i odd)

If v1 6 J v2 J vi J (for i even) v2k J

v1 or v2k is starting vertex of a blue edge v1 I2 \ I1 or v2k I2 \ I1

w.l.o.g. v1 J v1 I2 \ I1 ,{vi |i even} W2and {vi |i odd, 3} I1 I2

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

P := (v1, ..., v2k ) v1 J or v2k J

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

For {vi |i 3 and odd} I1 I2 every vertex isstarting vertex of a red and blue edge

Can interchange colors in P (rest in [W]unchanged)

one red edge more and all starting vertices inI1 {v1}

matching in [W] I1 {v1} partialtransversal of W

I1 {v1} I

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

I1 {v1} I

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

I1 {v1} I

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

I1 {v1} I

Proof

J W

j1

j2 W1

j3 W2

j4 W3

j5 W4

j6

Proof I3:

I1 {v1} I

Greedy Algorithm Transversal Matroids Edmonds Inte

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Greedy Algorithm And Edmonds Matroid Intersection Algorithmwilhelm/greedy-vortrag.pdf · Greedy Algorithm Transversal Matroids Edmonds’ Intersection Algorithm References Content