Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) groetschel@zib.de http://www.zib.de/groetschel

Independence Systems, Matroids,

the Greedy Algorithm, and related Polyhedra

Martin GrötschelSummary of Chapter 4

of the classPolyhedral Combinatorics (ADM III)

May 25, 2010

Matroids and Independence Systems

Let E be a finite set, I a subset of the power set of E.The pair (E,I ) is called independence system on E if the

following axioms are satisfied: (I.1) The empty set is in I. (I.2) If J is in I and I is a subset of J then I belongs to I.Let (E,I ) satisfy in addition: (I.3) If I and J are in I and if J is larger than I then there is an element j in J, j not in I, such that the union of I and j is in I.Then M=(E,I ) is called a matroid.

NotationLet (E,I ) be an independence system.Every set in I is called independent.Every subset of E not in I is called dependent.For every subset F of E, a basis of F is a subset of F that is independent and maximal with respect to this property.The rank r(F) of a subset F of E is the cardinality of a largest basis of F. The lower rank of F is the cardinality of a smallest basis of F.

ur (F)

The Largest Independent Set Problem

Problem:Let (E,I ) be an independence system with weights on theelements of E. Find an independent set of largest weight.

We may assume w.l.o.g. that all weights are nonnegative(or even positive), since deleting an element withnonpositive weight from an optimum solution, willnot decrease the value of the solution.

The Greedy AlgorithmLet (E,I ) be an independence system with weights c(e) on the elements of E. Find an independent set of largest weight.The Greedy Algorithm:1. Sort the elements of E such that2. Let 3. FOR i=1 TO n DO:

4. OUTPUT

1 2 ... 0.nc c cgreedyI : .

greedy greedy greedyIF I i THEN I := I i .I

greedyI .

A key idea is to interprete the greedy solution as the solution of a linear program.

Polytopes and LPsLet M=(E,I ) be an independence system with weights c(e) on the elements of E.

R

R

I I

( ) , 0

min s.t.

IND(M)

The LP relaxation

The dua

( ) ,

l

0 P

L

E

Ee e

e F

Te

e F

e

conv x I

conv x x r F F E x e E

c x x r F F E

x e E

FF E F e

min ( ) s.t. y ,

0

F e

F

y r F c e E

y F E

The Dual Greedy AlgorithmLet (E,I ) be an independence system with weights c(e) for all e.

After sorting the elements of E so thatset

1

i:= 1, 2, ..., i , i=1, 2, ..., n and: , i=1, 2

, ..., ny .

Ei iE ic c

1 2 1... 0, : 0n nc c c c

F E

FF e

min ( ),

s.t. y ,

0

F u

e

F

y r F

c e E

y F E

Then is a feasible solution of the dual LP

1y , i=1, 2, ..., niE i ic c (integral if the weights are integral))

ObservationLet (E,I ) be an independence system with weights c(e) for all e.

After sorting the elements of E so thatWe can express every greedy and optimum solution as follows:

1 2 1... 0, : 0n nc c c c

greedy 1 greedy1

opt 1 opt1

c(I ) ( ) I

c(I ) ( ) I

n

i i iin

i i ii

c c E

c c E

Rank QuotientLet (E,I ) be an independence system with weights c(e) for all e.

( ) 0

( ): min ) ( uF Er F

q r Fr F

The number q is between 0 and 1 and is called rank quotient of (E,I ).

Observation: q = 1 iff (E,I ) is a matroid.

The General Greedy Quality Guarantee

opt greed 1 11 1

1

eey gr dy

max ,s.t. ( ) , 0

max ,s.t. ( ) , 0 ,

c(I )

c(

inte

( ) ( )

( )

min

gral

I ( )I )

i

e e e ee E e F

e e

i u

e ee E e F

n n

i i i ii

n

E ui

ii

i

c x x r F F E x e E

c x x r F F E x e E

c c c c

y r E

x

E r E

y

FF E F e

FF E F e

q max ,s.t.

,s.t. y , 0

min ,s.

( )

t. y , 0

=

q max ,s.t. ( )

( )

q ( )

in , 0 ,

, 0

F e F

F e

e e e ee E e F

F

e e e ee E e

u

F

c x x r F F

c e E y F

E x e

E

y c e E y F E

c x x r F F E x e

r

E

F

F

x

E

r

opt q c(I

tegra

l

= ) a quality guaranteea quality guarantee

ConsequencesLet M=(E,I ) be an independence system with weights c(e) on the elements of E.

R

IIND(M) I

P(M) ( ) , 0

(a) P(M) = IND(M) if and only if M is a matroid(b) If M is a matroid then all optimum solutions of Theorem:

the primal LP max

Ee e

e F

T

conv x I

x x r F F E x e E

c x

FF E F e

s.t. ( ) , 0

are integral. If the weights are integral then the dual LP min ( ) s.t. y , 0

also has integral optimum solutions

e ee F

F e F

x r F F E x e E

y r F c e E y F E

t,

o

i ta.e., lly the sys dual inttem egis ral.

More Proofs Another proof of the completeness of the system

of nonnegativity constraints and rank inequalities will be given in the class on the blackboard, see further slides.

It will also be shown that a rank inequality x(F) ≤ r(F)

defines a facet of the matroid polytope if and only if the set F is closed and inseperable, seeMartin Grötschel, Facetten von Matroid-Polytopen, Operations Research Verfahren XXV, 1977, 306-313, downloadable fromhttp://www.zib.de/groetschel/pubnew/paper/groetschel1977d.pdf

A proof of the matroid intersection theorem will be given, see below.Martin

Grötschel

12

Completeness Proof of the Matroid Polytope

Martin Grötschel

13

Completeness Proof of the Matroid Polytope (continued)

Martin Grötschel

14

The proof above is from (GLS, pages 213-214), see http://www.zib.de/groetschel/pubnew/paper/groetschellovaszschrijver1988.pdf

The Forest Polytope

Martin Grötschel

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A Partition Matroid Polytope

Martin Grötschel

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The Branching and the Arborescence Polytope

Martin Grötschel

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The Branching and the Arborescence Polytope

Martin Grötschel

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The Matroid Intersection Polytope

Martin Grötschel

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The Matroid Intersection Polytope

Martin Grötschel

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The Matroid Intersection Polytope

Martin Grötschel

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The Matroid Intersection Polytope

Martin Grötschel

22

The proof above is from (GLS, pages 214-216), see http://www.zib.de/groetschel/pubnew/paper/groetschellovaszschrijver1988.pdf

Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) groetschel@zib.de http://www.zib.de/groetschel

Independence Systems, Matroids,

the Greedy Algorithm and related Polyhedra

Martin GrötschelSummary of Chapter 4

of the classPolyhedral Combinatorics (ADM III)

May 18, 2010The End