polyhedral optimization lecture 3 – part 2 m. pawan kumar pawan.kumar@ecp.fr slides available...
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Polyhedral OptimizationLecture 3 – Part 2
M. Pawan Kumar
pawan.kumar@ecp.fr
Slides available online http://cvn.ecp.fr/personnel/pawan/
• Operations on Matroids– Truncation– Deletion– Contraction– Duality of Deletion and Contraction
• Maximum Weight Independent Set
• Polytopes
Outline
t-Truncation of Matroid
M = (S, I) M’ = (S, I’)
X ∈I’ if two conditions are satisfied
(i) X ∈I
(ii) |X| ≤ t
M’ is a matroid Proof?
Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3
t = 2
Is {1, 2, 3} independent in M’ ?
NO
t-Truncation denoted by M’
Example – Uniform Matroid
S = {1, 2, 3, 4} k = 3
t = 2
Is {1, 2} independent in M’ ?
YES
t-Truncation denoted by M’
Example – Uniform Matroid
S = {1, 2, 3, 4} k = 3
t = 2
Is {1} independent in M’ ?
YES
t-Truncation denoted by M’
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
t = 3 t-Truncation denoted by M’
Independent in M’ ? NO
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
t = 3 t-Truncation denoted by M’
Independent in M’ ? NO
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
t = 3 t-Truncation denoted by M’
Independent in M’ ? YES
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
t = 3 t-Truncation denoted by M’
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
t = 3 t-Truncation denoted by M’
Independent in M’ ? NO
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
t = 3 t-Truncation denoted by M’
Independent in M’ ? NO
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
t = 3 t-Truncation denoted by M’
Independent in M’ ? YES
• Operations on Matroids– Truncation– Deletion– Contraction– Duality of Deletion and Contraction
• Maximum Weight Independent Set
• Polytopes
Outline
Deletion
M = (S, I) M\s = (S-s, I’)
X ∈I’ if two conditions are satisfied
(i) X S-s⊆
(ii) X ∈I
M\s is a matroid Proof?
Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3
s = 2
Is {1, 2, 3} independent in M’ ?
NO
Matroid after deletion of s denoted by M’
Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3
s = 2
Is {1, 3, 4, 5} independent in M’ ?
NO
Matroid after deletion of s denoted by M’
Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3
s = 2
Is {1, 3, 4} independent in M’ ?
YES
Matroid after deletion of s denoted by M’
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
s
Matroid after deletion of s denoted by M’
Independent in M’ ? NO
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
s
Matroid after deletion of s denoted by M’
Independent in M’ ? NO
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
s
Matroid after deletion of s denoted by M’
Independent in M’ ? YES
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after deletion of s denoted by M’
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after deletion of s denoted by M’
Independent in M’ ? NO
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after deletion of s denoted by M’
Independent in M’ ? NO
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after deletion of s denoted by M’
Independent in M’ ? YES
Effect on Rank Function
M = (S, I) M\s = (S-s, I’)
rM\s(X) = rM(X), for all X S-s⊆
Proof?
• Operations on Matroids– Truncation– Deletion– Contraction– Duality of Deletion and Contraction
• Maximum Weight Independent Set
• Polytopes
Outline
Contraction
M = (S, I) M/s = (S-s, I’)
X ∈I’ if two conditions are satisfied
(i) X S-s⊆
(ii) X + s ∈I
M/s is a matroid Proof?
Assume {s} ∈I
Contraction
M = (S, I) M/s = (S-s, I’)
X ∈I’ if two conditions are satisfied
(i) X S-s⊆
(ii) X ∈I
M/s is the same as M\s when {s} ∉ I
Assume {s} ∉ I
Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3
s = 2
Is {1, 2, 3} independent in M’ ?
NO
Matroid after contraction of s is M’
Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3
s = 2
Is {1, 3, 4} independent in M’ ?
NO
Matroid after contraction of s is M’
Example – Uniform Matroid
S = {1, 2, 3, 4, 5, 6} k = 3
s = 2
Is {1, 4} independent in M’ ?
YES
Matroid after contraction of s is M’
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
s
Matroid after contraction of s denoted by M’
Independent in M’ ? NO
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
s
Matroid after contraction of s denoted by M’
Independent in M’ ? NO
Example – Linear Matroid
1
2
3
4
2
4
6
8
1
1
1
1
2
2
2
2
3
3
3
3
1
2
1
2
2
4
2
4
1
2
1
2
2
4
2
4
s
Matroid after contraction of s denoted by M’
Independent in M’ ? YES
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after contraction of s denoted by M’
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after contraction of s denoted by M’
Independent in M’ ? NO
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after contraction of s denoted by M’
Independent in M’ ? NO
Example – Graphic Matroid
v1
v0
v2
v6
v4
v5
v3
s
Matroid after contraction of s denoted by M’
Independent in M’ ? YES
Effect on Rank Function
M = (S, I) M/s = (S-s, I’)
rM/s(X) = rM(X+s) – rM({s}), for all X S-s⊆
Proof?
• Operations on Matroids– Truncation– Deletion– Contraction– Duality of Deletion and Contraction
• Maximum Weight Independent Set
• Polytopes
Outline
Duality
(M/s)* = M*\s Proof?
If {s} ∉ I, deletion doesn’t affect independence
If {s} ∉ I, contraction doesn’t affect independence
Proof is trivial
Duality
(M/s)* = M*\s Proof? Assume {s} ∈I
X is independent in (M/s)*
⟺ X B⊆ 1 for some base B1 of (M/s)*
⟺ X ∩ B2 = Null for some base B2 of M/s
s X∉
⟺ X ∩ B3 = Null for some base B3 = B2 + s of M
⟺ X B⊆ 4 for some base B4 of M*
Hence proved.
• Operations on Matroids
• Maximum Weight Independent Set
• Polytopes
Outline
Independent Set
Matroid M = (S, I)
X S is independent if X ⊆ I
X S is dependent if X ⊆ ∉ I
Weight of an Independent Set
Matroid M = (S, I)
Weight function w: S → Non-negative Real
Weight of an independent set X
The sum of weight of its elements
Weight of an Independent Set
Matroid M = (S, I)
w(X) = ∑s X∈ w(s)
Weight of an independent set X
Weight function w: S → Non-negative Real
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
X = {1, 3, 5}
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
X = {1, 3, 5}
w(X)?
15
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Forest X
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Forest X
w(X)?
10
Maximum Weight Independent Set
Matroid M = (S, I)
maxX∈I w(X)
Find an independent set with maximum weight
Weight function w: S → Non-negative Real
Maximum Weight Independent Set
Matroid M = (S, I)
maxX∈I ∑s X∈ w(s)
Find an independent set with maximum weight
Weight function w: S → Non-negative Real
True or False
There exists an optimal solutionthat is a base of the matroid
TRUE
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Feasible Solutions?
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Optimal Solution?
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Feasible Solutions?
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Optimal Solution?
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Efficient Algorithm?
• Operations on Matroids
• Maximum Weight Independent Set– Greedy Algorithm– Efficiency– Optimality: Necessity– Optimality: Sufficiency– Extensions
• Polytopes
Outline
Greedy Algorithm
Start with an empty set
Repeat
Pick a new element with maximum weight
such that the new set is independent
Until no more elements can be added
Add the element to the set
Greedy Algorithm
X ← ϕ
Repeat
Pick a new element with maximum weight
such that the new set is independent
Until no more elements can be added
Add the element to the set
Greedy Algorithm
X ← ϕ
Repeat
s* = argmaxx S\X∈ w(s)
such that the new set is independent
Until no more elements can be added
Add the element to the set
Greedy Algorithm
X ← ϕ
Repeat
s* = argmaxx S\X∈ w(s)
such that X {s} ∪ ∈ I
Until no more elements can be added
Add the element to the set
Greedy Algorithm
X ← ϕ
Repeat
s* = argmaxx S\X∈ w(s)
such that X {s} ∪ ∈ I
Until no more elements can be added
X ← X {s}∪
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
2 6
2
5
31
3 2
4
S = E
• Operations on Matroids
• Maximum Weight Independent Set– Greedy Algorithm– Efficiency– Optimality: Necessity– Optimality: Sufficiency– Extensions
• Polytopes
Outline
Efficiency of Greedy Algorithm
X ← ϕ
Repeat
s* = argmaxx S\X∈ w(s)
such that X {s} ∪ ∈ I
Until no more elements can be added
X ← X {s}∪
O(|S|)
O(|S|)
Independence Testing Oracle
Efficiency depends on independence testing
Uniform matroid?
Graphic matroid?
Brute-force is not an option
• Operations on Matroids
• Maximum Weight Independent Set– Greedy Algorithm– Efficiency– Optimality: Necessity– Optimality: Sufficiency– Extensions
• Polytopes
Outline
Optimality: Necessity
(S, I) is a matroid
⟹(S, I) admits an optimal greedy algorithm
Proof?
Proof Sketch
Matroid M = (S, I)
Proof by induction
Current solution X
X is part of optimal solution B (base of M)
Proof Sketch
Next step chooses to add s If s B, trivial∈
Otherwise, consider base B’ such that
B’ = (B {s})\{t}∪
w(B’) ≥ w(B) by construction
X {s} B’ ∪ ⊆ ⊆ B {s}∪
• Operations on Matroids
• Maximum Weight Independent Set– Greedy Algorithm– Efficiency– Optimality: Necessity– Optimality: Sufficiency– Extensions
• Polytopes
Outline
Optimality: Sufficiency
(S, I) is a matroid
⟹(S, I) admits an optimal greedy algorithm
Proof?
Proof Sketch
X ∈ I
Proof by contradiction
There should exist an s Y\X, X {s} ∈ ∪ ∈ I
Matroid M = (S, I)
Y ∈ I k = |X| < |Y|
Proof Sketch
w(s) =
k+2, if s X ∈
k+1, if s Y\X ∈
0, otherwise
Greedy solution has weight k(k+2)
w(Y) ≥ (k+1)(k+1) > k(k+2)
Recall k = |X|
• Operations on Matroids
• Maximum Weight Independent Set– Greedy Algorithm– Efficiency– Optimality: Necessity– Optimality: Sufficiency– Extensions
• Polytopes
Outline
Maximum Weight Base
Matroid M = (S, I)
maxX∈B w(X)
Find a base with maximum weight
Does greedy work? YES
Weight function w: S → Real
Minimum Weight Base
Matroid M = (S, I)
minX∈B w(X)
Find a base with minimum weight
Does greedy work? YES
Weight function w: S → Real
Maximum Weight Independent Set
Matroid M = (S, I)
maxX∈I w(X)
Find an independent set with maximum weight
Does greedy work? YES
Weight function w: S → Real
• Operations on Matroids
• Maximum Weight Independent Set
• Polytopes
Outline
Incidence Vector of Set
Matroid M = (S, I) Set X S⊆
Incidence vector vX {0,1}∈ |S|
vX(s) =
1, if s X ∈
0, if s X ∉
Example (Uniform Matroid)
S = {1,2,…,9} s w(s)
1 10
2 5
3 2
4 1
5 3
6 6
7 12
8 2
9 1
k = 4
X = {1, 3, 5}
1
0
1
0
1
0
0
0
0
vX?
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
e1 e2
e4
e6
e5e3
e7 e9
e8
S = E
X S⊆
Example (Graphic Matroid)
v1
v0
v2
v6
v4
v5
v3
S = E
X S ⊆
vX?
e1 e2
e4
e6
e5e3
e7 e9
e8
1
0
0
1
0
0
0
1
1
Incidence Vectors of Independent Sets
Convex HullAx ≤ b
Independent Set Polytope
A?
b?
Matroid M = (S, I)
First, a property !!
vX {0,1}∈ |S|, X ∈ I
Maximum Weight Independent Set
w* = maxX∈I w(X) Matroid M = (S, I)
S = {s1,s2,…,sm}, such that w(si) ≥ w(si+1) ≥ 0
Ui = {s1,…si}
X = {si | rM(Ui) > rM(Ui-1)}
Optimal solution Why?
Maximum Weight Independent Set
w* = maxX∈I w(X) Matroid M = (S, I)
w(X)
X = {si | rM(Ui) > rM(Ui-1)}
= ∑s X∈ w(s)
= ∑i w(si)(rM(Ui)-rM(Ui-1))
= ∑i λi rM(Ui)
w = ∑i λi vUiRelationship between w and λ?
Maximum Weight Independent Set
w* = maxX∈I w(X) Matroid M = (S, I)
w(X) = w* = ∑i λi rM(Ui)
such that w = ∑i λi vUi
We need to remember this for what follows
• Operations on Matroids
• Maximum Weight Independent Set
• Polytopes– Independent Set Polytope– Base Polytope– Spanning Set Polytope
Outline
Independent Set Polytope
Convex HullAx ≤ b
Matroid M = (S, I)
vX {0,1}∈ |S|, X ∈ I
x Real∈ |S|
Independent Set Polytope
vX {0,1}∈ |S|, X ∈ I
Matroid M = (S, I)
x Real∈ |S|
xs ≥ 0, for all s S ∈
∑s U∈ xs ≤ rM(U), for all U S ⊆
Necessary conditions
Why?
Why?
Sufficient for integral x Proof?
Independent Set Polytope
vX {0,1}∈ |S|, X ∈ I
Matroid M = (S, I)
x Real∈ |S|
xs ≥ 0, for all s S ∈
∑s U∈ xs ≤ rM(U), for all U S ⊆
Necessary conditions
Why?
Why?
Sufficient for all x Proof?
Independent Set Polytope
maxx wTx
xs ≥ 0, for all s S ∈
∑s U∈ xs ≤ rM(U), for all U S ⊆
Dual?
miny ∑U S⊆ yUrM(U)
yU ≥ 0, for all U S ⊆
∑U S⊆ yUvU ≥ w
Independent Set Polytope
maxx wTx
xs ≥ 0, for all s S ∈
∑s U∈ xs ≤ rM(U), for all U S ⊆
miny ∑U S⊆ yUrM(U)
yU ≥ 0, for all U S ⊆
∑U S⊆ yUvU = w
Greedy algorithm integral solution
x’
Independent Set Polytope
maxx wTx
xs ≥ 0, for all s S ∈
∑s U∈ xs ≤ rM(U), for all U S ⊆
miny ∑U S⊆ yUrM(U)
yU ≥ 0, for all U S ⊆
∑U S⊆ yUvU = w
Solution obtained using the property
x’
y’
Independent Set Polytope
maxx wTx
xs ≥ 0, for all s S ∈
∑s U∈ xs ≤ rM(U), for all U S ⊆
miny ∑U S⊆ yUrM(U)
yU ≥ 0, for all U S ⊆
∑U S⊆ yUvU = w
Relationship between objectives for x’ and y’ ?
x’
y’
Independent Set Polytope
∑s w(s)x’s = ∑U S⊆ y’UrM(U)
Dual upper bounds primal
x’ and y’ must be optimal solutions
For all integer w, we have integer optimal value
Hence provedNegative values in w?
• Operations on Matroids
• Maximum Weight Independent Set
• Polytopes– Independent Set Polytope– Base Polytope– Spanning Set Polytope
Outline
Base Polytope
Convex HullAx ≤ b
Matroid M = (S, I)
vX {0,1}∈ |S|, X ∈ B
x Real∈ |S|
Base Polytope
vX {0,1}∈ |S|, X ∈ B
Matroid M = (S, I)
x Real∈ |S|
xs ≥ 0, for all s S ∈
∑s U∈ xs ≤ rM(U), for all U S ⊆
∑s S∈ xs = rM(S)
• Operations on Matroids
• Maximum Weight Independent Set
• Polytopes– Independent Set Polytope– Base Polytope– Spanning Set Polytope
Outline
Spanning Set Polytope
Convex HullAx ≤ b
Matroid M = (S, I)
vX {0,1}∈ |S|, X ∈ S
x Real∈ |S|
Independent set of M*
S\X?
Spanning Set Polytope
vX {0,1}∈ |S|, X ∈ B
Matroid M = (S, I)
x Real∈ |S|
1-xs ≥ 0, for all s S ∈
∑s U∈ (1-xs) ≤ rM*(U), for all U S ⊆
xs ≥ 0, for all s S ∈
Spanning Set Polytope
vX {0,1}∈ |S|, X ∈ B
Matroid M = (S, I)
x Real∈ |S|
xs ≤ 1, for all s S ∈
∑s U∈ (1-xs) ≤ rM*(U), for all U S ⊆
xs ≥ 0, for all s S ∈
Spanning Set Polytope
vX {0,1}∈ |S|, X ∈ B
Matroid M = (S, I)
x Real∈ |S|
xs ≤ 1, for all s S ∈
∑s U∈ xs ≥ rM(S) - rM(S\U), for all U S ⊆
xs ≥ 0, for all s S ∈
Questions?
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