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 &June 1, 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. Important property, submodularity: The lower rank of F is the cardinality of a smallest basis of F.

ur (F) r(S T)+r(S T) r(S)+r(T) S,T E

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 c

greedyI : .

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.

The greedy algorithm works for matroids

Proof using axiom (I.3) on the blackboard.

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

I I

( ) , 0

max s.t.

IND(M)

The LP relaxation

The dual LP

( ) ,

0

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

R

FF E F e

min ( ) s.t. y ,

0

F e

F

y r F c e E

y F E

P(M) ( ) , 0 E

e ee F

x x r F F E x e ER

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 .

E

i 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 by construction.

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 ii

n

i i ii

c c E

c c E

Rank Quotient

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

( ) 0

( ): min

)

( u

F Er F

qr 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

( ) 0

( ): min

)

( u

F Er F

qr Fr F

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

IIND( ) I

P( ) ( ) , 0

(a) P( ) = IND( ) if and only if M is a matroid

(b) If M is a matroid then all optimum solution

Theor

s of

em:

the primal LP

max

Ee e

e F

T

I conv x I

I x x r F F E x e E

I I

c x

R

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.

Consequences

Theorem. For every independence system (E,I with weights

c(e) for all elements e of E,

max cTx, xϵP(I) ≥ c(Iopt) ≥ c(Igreedy) ≥ q max cTx, xϵP(I) ≥ q

c(Iopt),

in other words, the greedy solution value is bounded from below by q times the maximum value of the LP relaxation and not only by q times the optimum value of the weighted independent set problem.

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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.

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Completeness Proof of the Matroid Polytope

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Completeness Proof of the Matroid Polytope (continued)

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

Facets of the matroid polytope

It will also be shown on the blackboard 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 inseparable, seeMartin Grötschel, Facetten von Matroid-Polytopen, Operations Research Verfahren XXV, 1977, 306-313, downloadable fromhttp://www.zib.de/groetschel/pubnew/paper/groetschel1977d.pdf

A subset F of E is closed if r(F U {e})>r(F) for all e in E\F.

A subset F of E is separable if there exist two nonempty disjoint subsets F1 and F2 of F whose union is F and such that

r(F)= r(F1)+ r(F2).Martin Grötschel

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The Forest Polytope

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A Partition Matroid Polytope

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The 1-Tree Polytope1-trees come up as relaxations of the symmetric travelling salesman problem. Given a complete graph Kn=(V,E), V={1,2,

…,n}. A 1-tree is the union of the edge set of a spanning tree of the complete graph on the node set {2,3,…,n} and two edges with endnode 1. Every 1-tree has n edges and contains exactly one cycle. This means that every travelling salesman tour is a 1-tree. The set of 1-trees is the set of bases of a matroid on E. A complete description of the convex hull of the incidence vectors of all 1-trees in Kn is given by:

0≤ xe ≤ 1 for all e in E

x(E(W)) ≤ |W| - 1 for all node sets W in V not containing 1

x(δ(1)) = 2

x(E) = nMartin Grötschel

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The Branching and the Arborescence Polytope

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The Branching and the Arborescence Polytope

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The Matroid Intersection Polytope

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The Matroid Intersection Polytope

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The Matroid Intersection Polytope

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The Matroid Intersection Polytope

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

Submodular Functions and Polymatroids

The whole polyhedral and algorithmic theory developed so far can be generalized to submodular functions and polymatroids.

This is worked out in detail in Chapter 10 of (GLS), seehttp://www.zib.de/groetschel/pubnew/paper/groetschellovaszschrijver1988.pdf

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Claude Berge (perfect graphs) andJack Edmonds (matching and matroids)

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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