OXFORD UNIVERSITY
COMPUTING LABORATORY
The Expressive Power of Binary Submodular
Functions
Stanislav Živný, David Cohen, Peter Jeavons
Computing Laboratory, University of Oxford
Rutgers, 22 January 2009
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by (or decomposed into) quadratic submodular
polynomials?
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by (or decomposed into) quadratic submodular
polynomials?
Not all.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
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Motivation
Expressible by quadratic submodular polynomials
= minimisation via (s, t)-Min-Cut.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
pseudo-Boolean: domain {0, 1}, coefficients in R
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Example
Let I ⊆ [n], |I| = k, and
p(x1 , . . . , xn ) = −
Y
xi .
i∈I
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Example
Let I ⊆ [n], |I| = k, and
p(x1 , . . . , xn ) = −
Y
xi .
i∈I
Then,
p(x1 , . . . , xn ) = min {−y + y
y∈{0,1}
Živný et al.
X
(1 − xi )}.
i∈I
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity
p(x1 , . . . , xn ) = a0 +
n
X
i=1
Živný et al.
ai xi +
X
1≤i<j≤n
The Expressive Power of Binary Submodular Functions
aij xi xj
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity
p(x1 , . . . , xn ) = a0 +
n
X
i=1
ai xi +
X
aij xi xj
1≤i<j≤n
Definition
(Quadratic) p is submodular ⇔ aij ≤ 0 for all 1 ≤ i < j ≤ n.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
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Submodularity, cont’d
p(x1 , . . . , pn ) =a0 +
n
X
ai xi +
i=1
X
X
aij xi xj +
1≤i<j≤n
aijk xi xj xk + . . . +
1≤i<j<k≤n
Živný et al.
The Expressive Power of Binary Submodular Functions
X
|I|=k
aI xI
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity, cont’d
p(x1 , . . . , pn ) =a0 +
n
X
ai xi +
i=1
X
X
aij xi xj +
1≤i<j≤n
aijk xi xj xk + . . . +
1≤i<j<k≤n
X
aI xI
|I|=k
Definition
p is submodular ⇔ δij (x) ≤ 0, where
δ1,2 (x) = p(1, 1, x) − p(1, 0, x) − p(0, 1, x) + p(0, 0, x) is the
second-order derivate of p.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity, cont’d
• key concept in combinatorial optimisation
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity, cont’d
• key concept in combinatorial optimisation
• discrete analogue of convexity
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity, cont’d
• key concept in combinatorial optimisation
• discrete analogue of convexity
• given finite V and ψ : 2V → R,
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity, cont’d
• key concept in combinatorial optimisation
• discrete analogue of convexity
• given finite V and ψ : 2V → R,
ψ(A ∪ B) + ψ(A ∩ B) ≤ ψ(A) + ψ(B) for all A, B ⊆ V
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Submodularity, cont’d
• key concept in combinatorial optimisation
• discrete analogue of convexity
• given finite V and ψ : 2V → R,
ψ(A ∪ B) + ψ(A ∩ B) ≤ ψ(A) + ψ(B) for all A, B ⊆ V
• examples: cut fns, matroid rank fns, entropy fns
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Expressibility
Definition
p ∈ hLi ⇔ p(x) = min
z
Živný et al.
X
pi (x, z) + c, where pi ∈ L, c ∈ R.
i
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Example
Let I ⊆ [n], |I| = k, and
p(x1 , . . . , xn ) = −
Y
xi .
i∈I
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Example
Let I ⊆ [n], |I| = k, and
p(x1 , . . . , xn ) = −
Y
xi .
i∈I
Then,
p(x1 , . . . , xn ) = min {−y + y
y∈{0,1}
Živný et al.
X
(1 − xi )}.
i∈I
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Another example
p(x1 , x2 , x3 , x4 ) = x1 x2 x3 x4 − x1 x2 − x1 x3 x4 − x2 x3 x4
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Another example
p(x1 , x2 , x3 , x4 ) = x1 x2 x3 x4 − x1 x2 − x1 x3 x4 − x2 x3 x4
p(x1 , x2 , x3 , x4 ) = min (y1 + 2y2 − y1 y2 − y1 x1 − y1 x2 − y2 x3 − y2 x4 )
y1 ,y2
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Another example
p(x1 , x2 , x3 , x4 ) = x1 x2 x3 x4 − x1 x2 − x1 x3 x4 − x2 x3 x4
p(x1 , x2 , x3 , x4 ) = min (y1 + 2y2 − y1 y2 − y1 x1 − y1 x2 − y2 x3 − y2 x4 )
y1 ,y2
new: 2 extra variables, 1 is provably not enough
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Yet another example
p(x1 , x2 , x3 , x4 ) = − x1 x2 x3 x4 + x1 x3 x4 + x2 x3 x4
− x1 x3 − x1 x4 − x2 x3 − x2 x4 − x3 x4
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Yet another example
p(x1 , x2 , x3 , x4 ) = − x1 x2 x3 x4 + x1 x3 x4 + x2 x3 x4
− x1 x3 − x1 x4 − x2 x3 − x2 x4 − x3 x4
new: not expressible!
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Problem
Which submodular polynomials can be expressed
by quadratic submodular polynomials?
Expressible by quadratic submodular polynomials
= minimisation via (s, t)-Min-Cut.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of polynomials
• minimisation of polynomials [Boros & Hammer ’02]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of polynomials
• minimisation of polynomials [Boros & Hammer ’02]
• minimisation of quadratic polynomials [Rosenberg ’75]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of polynomials
• minimisation of polynomials [Boros & Hammer ’02]
• minimisation of quadratic polynomials [Rosenberg ’75]
τ (G) = minimum vertex cover of G = (V, E)
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of polynomials
• minimisation of polynomials [Boros & Hammer ’02]
• minimisation of quadratic polynomials [Rosenberg ’75]
τ (G) = minimum
cover of G = (V, E)
P vertexP
τ (G) = min( i∈V xi + (i,j)∈E (1 − xi )(1 − xj ))
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of polynomials
• minimisation of polynomials [Boros & Hammer ’02]
• minimisation of quadratic polynomials [Rosenberg ’75]
τ (G) = minimum
cover of G = (V, E)
P vertexP
τ (G) = min( i∈V xi + (i,j)∈E (1 − xi )(1 − xj ))
• minimisation of submodular polynomials
[Schrijver ’00, Iwata & Fleischer & Fujishige ’01, Orlin ’07]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of polynomials
• minimisation of polynomials [Boros & Hammer ’02]
• minimisation of quadratic polynomials [Rosenberg ’75]
τ (G) = minimum
cover of G = (V, E)
P vertexP
τ (G) = min( i∈V xi + (i,j)∈E (1 − xi )(1 − xj ))
• minimisation of submodular polynomials
[Schrijver ’00, Iwata & Fleischer & Fujishige ’01, Orlin ’07]
• minimisation of quadratic submodular polynomials
= (s, t)-Min-Cut [Hammer ’65]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Reduction to (s, t)-Min-Cut
If
p(x1 , . . . , xn ) = a0 +
n
X
i=1
ai xi +
X
1≤i<j≤n
where aij ≤ 0 for all 1 ≤ i < j ≤ n.
Živný et al.
The Expressive Power of Binary Submodular Functions
aij xi xj ,
OXFORD UNIVERSITY
COMPUTING LABORATORY
Reduction to (s, t)-Min-Cut
If
p(x1 , . . . , xn ) = a0 +
n
X
ai xi +
i=1
where aij ≤ 0 for all 1 ≤ i < j ≤ n.
Then,
X
X
p = a00 +
a0i xi +
a0j (1 − xj ) +
i∈P
j∈N
X
X
1≤i<j≤n
s.t. a0i , a0j , a0ij ≥ 0.
Živný et al.
aij xi xj ,
1≤i<j≤n
The Expressive Power of Binary Submodular Functions
a0ij (1 − xi )xj
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COMPUTING LABORATORY
Reduction to (s, t)-Min-Cut, cont’d
P
a0i
a0ik
s
a0k
Živný et al.
xi
xk
N
a0il
a0ij
a0kl
a0jk
xl
a0l
a0jl
xj
The Expressive Power of Binary Submodular Functions
t
a0j
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of submodular polynomials
• quadratic in O(n3 ) using (s, t)-Min-Cut [Hammer ’65]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of submodular polynomials
• quadratic in O(n3 ) using (s, t)-Min-Cut [Hammer ’65]
• degree k in O(n6 + n5 L), L is the evaluation time [Orlin ’07]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of submodular polynomials
• quadratic in O(n3 ) using (s, t)-Min-Cut [Hammer ’65]
• degree k in O(n6 + n5 L), L is the evaluation time [Orlin ’07]
• cubic expressible by quadratic (1 new variable per cubic term)
⇒ O(n3 ) using (s, t)-Min-Cut [Billionet & Minoux ’85]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of submodular polynomials
• quadratic in O(n3 ) using (s, t)-Min-Cut [Hammer ’65]
• degree k in O(n6 + n5 L), L is the evaluation time [Orlin ’07]
• cubic expressible by quadratic (1 new variable per cubic term)
⇒ O(n3 ) using (s, t)-Min-Cut [Billionet & Minoux ’85]
• 0/1-valued polynomials [Creignou, Khanna & Sudan ’01]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of submodular polynomials
• quadratic in O(n3 ) using (s, t)-Min-Cut [Hammer ’65]
• degree k in O(n6 + n5 L), L is the evaluation time [Orlin ’07]
• cubic expressible by quadratic (1 new variable per cubic term)
⇒ O(n3 ) using (s, t)-Min-Cut [Billionet & Minoux ’85]
• 0/1-valued polynomials [Creignou, Khanna & Sudan ’01]
• testing submodularity of degree 4 is co-NP-complete
[Gallo & Simeone ’88]
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Minimisation of submodular polynomials
• quadratic in O(n3 ) using (s, t)-Min-Cut [Hammer ’65]
• degree k in O(n6 + n5 L), L is the evaluation time [Orlin ’07]
• cubic expressible by quadratic (1 new variable per cubic term)
⇒ O(n3 ) using (s, t)-Min-Cut [Billionet & Minoux ’85]
• 0/1-valued polynomials [Creignou, Khanna & Sudan ’01]
• testing submodularity of degree 4 is co-NP-complete
[Gallo & Simeone ’88]
• here we assume submodularity (CSP, CV)
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Result: Expressibility of fans
Theorem [Ž., Cohen, Jeavons ’08]
All FANS are expressible by quadratic submodular polynomials.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Result: Expressibility of fans
Theorem [Ž., Cohen, Jeavons ’08]
All FANS are expressible by quadratic submodular polynomials.
(generalise previous expressibility results)
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Result: Expressibility of fans
Theorem [Ž., Cohen, Jeavons ’08]
All FANS are expressible by quadratic submodular polynomials.
(generalise previous expressibility results)
(arity k fan expressible with O(k) extra variables)
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
0
1234
0
0
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0
0
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0
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0
0
0
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1
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0
∅
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
0
1234
0
0
0
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−1
−1
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−1
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∅
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
0
1234
−2
0
0
0
123
124
134
234
−1
−1
0
−1
0
0
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0
0
0
0
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0
∅
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
0
1234
−2
−1
−1
−1
123
124
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234
−1
−1
0
−1
0
0
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0
0
0
0
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∅
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
−2
1234
−2
−1
−1
−1
123
124
134
234
−1
−1
0
−1
0
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∅
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
More on expressibility
Theorem [Cooper, Cohen, Jeavons ’06]
1
p ∈ hLi ⇔ fPol(L) ⊆ fPol({p})
2
p ∈ hLi ⇒ p ∈ hLi with at most 22 extra variables
Živný et al.
k
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Result: Non-expressibility
Theorem [Ž., Cohen, Jeavons ’08]
Let p ∈ Lsub,4 . Then the following are equivalent:
1
p ∈ hLsub,2 i
2
p ∈ Cone(Lf ans,4 )
3
Fsep ∈ fPol({p})
4
∀{i, j}, {k, l} ⊆ [n] distinct : aij + akl + aijk + aijl ≤ 0.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
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Result: Non-expressibility
Theorem [Ž., Cohen, Jeavons ’08]
Let p ∈ Lsub,4 . Then the following are equivalent:
1
p ∈ hLsub,2 i
2
p ∈ Cone(Lf ans,4 )
3
Fsep ∈ fPol({p})
4
∀{i, j}, {k, l} ⊆ [n] distinct : aij + akl + aijk + aijl ≤ 0.
Proof: (2) ⇒ (1) (Fans expressible), (1) ⇒ (3) (Characterisation
of fPol), (3) ⇒ (2) (The same polyhedra), (3) ⇔ (4)
Živný et al.
The Expressive Power of Binary Submodular Functions
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Results, cont’d
Theorem [Ž., Cohen, Jeavons ’08]
• Canonical examples qin.
• Testing whether p ∈ Lsub,4 ∩ hLsub,2 i is co-NP-complete.
• Not all extreme rays are Fans (Promislow & Young).
• Optimal number of variables.
Živný et al.
The Expressive Power of Binary Submodular Functions
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Results, cont’d
Theorem [Ž., Cohen, Jeavons ’08]
• Canonical examples qin.
• Testing whether p ∈ Lsub,4 ∩ hLsub,2 i is co-NP-complete.
• Not all extreme rays are Fans (Promislow & Young).
• Optimal number of variables.
• KEY: fPol(Lsub,2 ) are conservative Hamming distance
non-increasing weighted mappings.
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
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VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
Živný et al.
The Expressive Power of Binary Submodular Functions
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VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
• (finite) Variables, (finite) Domain, Constraints
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
• (finite) Variables, (finite) Domain, Constraints
• constraint = cost function: scope → Q+ ∪ {∞}
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
• (finite) Variables, (finite) Domain, Constraints
• constraint = cost function: scope → Q+ ∪ {∞}
• goal: minimise the total cost
(=objective function=sum of all constraints)
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
• (finite) Variables, (finite) Domain, Constraints
• constraint = cost function: scope → Q+ ∪ {∞}
• goal: minimise the total cost
(=objective function=sum of all constraints)
• CSP: hard constraints only ({0, ∞})
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
• (finite) Variables, (finite) Domain, Constraints
• constraint = cost function: scope → Q+ ∪ {∞}
• goal: minimise the total cost
(=objective function=sum of all constraints)
• CSP: hard constraints only ({0, ∞})
• constraints represented explicitly (tables) ⇒ bound on arity
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
• (finite) Variables, (finite) Domain, Constraints
• constraint = cost function: scope → Q+ ∪ {∞}
• goal: minimise the total cost
(=objective function=sum of all constraints)
• CSP: hard constraints only ({0, ∞})
• constraints represented explicitly (tables) ⇒ bound on arity
• cost function = polynomial
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
VCSP
• VALUED C ONSTRAINT S ATISFACTION P ROBLEM
• VCSP = hV, D, Ci
• (finite) Variables, (finite) Domain, Constraints
• constraint = cost function: scope → Q+ ∪ {∞}
• goal: minimise the total cost
(=objective function=sum of all constraints)
• CSP: hard constraints only ({0, ∞})
• constraints represented explicitly (tables) ⇒ bound on arity
• cost function = polynomial
• VCSP instance ⇒ p =
Živný et al.
P
i pi
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
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Conclusions
• not all submodular polynomials are expressible over Lsub,2
• most submodular submodular polynomials are not
• extreme rays vs. expressibility
Živný et al.
The Expressive Power of Binary Submodular Functions
OXFORD UNIVERSITY
COMPUTING LABORATORY
Q&A
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Many thanks to the organisers!
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http://zivny.cz/
Živný et al.
The Expressive Power of Binary Submodular Functions