independence systems, matroids, the greedy algorithm, and related polyhedra
DESCRIPTION
Independence Systems, Matroids, the Greedy Algorithm, and related Polyhedra. Martin Grötschel Summary of Chapter 4 of the class Polyhedral 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. - PowerPoint PPT PresentationTRANSCRIPT
Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] 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.
Martin Grötschel
6
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) [email protected] 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