Discrete Convexity and Polynomial Solvability in ... - Semantic Scholar
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Discrete Convexity and Polynomial Solvability in Minimum 0-Extension Problems Hiroshi HIRAI The University of Tokyo [email protected]
RIMS, Kyoto, October 17, 2012
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1 / 27
. Contents Minimum 0-extension problems Known results on P/NP-hard classification Main result Proof idea from Discrete Convex Analysis & Valued-CSP Some technical detail . Notation . (semi)metric d on X ⇔ d : X × X → R+ , d(x, x) = 0, .
d(x, y ) = d(y , x),
d(x, z)+d(z, y ) ≥ d(x, y ).
(X , d): (semi)metric space
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.
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2 / 27
. Minimum 0-Extension Problem (Karzanov 1998) (S, µ): finite metric space
Def: extension of (S, µ) ⇔ metric space (X , d) s.t. X ⊇ S and d|S = µ Def: 0-extension of (S, µ) ⇔ extension (X , d) s.t. ∃ρ : X → S: ρ|S = id and d = µ ◦ ρ (X, d)
(X, d)
µ(u, s)
µ(s, t)
s u
t (S, µ)
ρ µ(u, t)
extension
0-extension
.
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.
.
3 / 27
. Minimum 0-Extension Problem (Karzanov 1998) (S, µ): finite metric space Def: extension of (S, µ) ⇔ metric space (X , d) s.t. X ⊇ S and d|S = µ Def: 0-extension of (S, µ) ⇔ extension (X , d) s.t. ∃ρ : X → S: ρ|S = id and d = µ ◦ ρ . Minimum 0-Extension Problem on (S, µ) . ( ) Given a finite set X ⊇ S and c : X2 → Q+ , minimize
∑
c(xy )d(x, y )
xy
.
subject to
(X , d): 0-extension of (S, µ)
.
.
.
.
.
.
3 / 27
. 0-Ext[µ]
Minimize
∑
c(xy )µ(ρ(x), ρ(y ))
xy
.
subject to
ρ : X → S, ρ|S = id.
.
.
.
.
.
.
4 / 27
. 0-Ext[µ]
Minimize
∑
c(xy )µ(ρ(x), ρ(y ))
xy
subject to
.
ρ : X → S, ρ|S = id.
. → MRF optimization form: ∑∑ ∑ Minimize ciq µ(q, ρi ) + cij µ(ρi , ρj ) i
.
subject to
q∈S
i<j
(ρ1 , ρ2 , . . . , ρn ) ∈ S × S × · · · × S
Applications: computer vision, clustering, learning theory,...
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.
.
.
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.
4 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
.
subject to
ρ : X → S, ρ|S = id.
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.
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.
5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
.
subject to
ρ : X → S, ρ|S = id.
s
Γ = K2
t
X
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5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ c(xy )dΓ (ρ(x), ρ(y )) 0-Ext[Γ ] Minimize xy
.
subject to
ρ : X → S, ρ|S = id.
Minimum cut s
Γ = K2 1
t
X
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5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
.
subject to
ρ : X → S, ρ|S = id.
Γ = K
X .
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5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ c(xy )dΓ (ρ(x), ρ(y )) 0-Ext[Γ ] Minimize xy
.
subject to
ρ : X → S, ρ|S = id.
Multi-terminal cut
Γ = K
1
X .
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5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
subject to
.
ρ : X → S, ρ|S = id.
Γ 2 1
X .
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.
.
.
5 / 27
. Tractability Question (Karzanov 1998, 2004)
Γ = K2 ⇒ minimum cut ⇒ P Γ = Kn ⇒ multi-terminal cut ⇒ NP-hard . Question . What is Γ for which 0-Ext[Γ ] is in P ? .
(n ≥ 3)
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.
.
.
.
.
6 / 27
. A classical result in location theory . Theorem (Picard-Ratliff 1978) . If . Γ is a tree, then 0-Ext[Γ ] is in P.
.
.
.
.
.
.
7 / 27
. A classical result in location theory . Theorem (Picard-Ratliff 1978) . If . Γ is a tree, then 0-Ext[Γ ] is in P. Obs. Γ ∈ P and Γ ′ ∈ P ⇒ Γ × Γ ′ ∈ P (× := Cartesian product)
Γ
Γ′
Γ × Γ′ .
.
.
.
.
.
7 / 27
. Median graph .
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median. u
y x
y
x z w z
v
.
.
.
.
.
.
8 / 27
. Median graph .
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median.
c.f. Median graph ≃ graph of CAT(0) cube complex (Chepoi 2000) .
.
.
.
.
.
8 / 27
. Median graph .
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median.
median
c.f. Median graph ≃ graph of CAT(0) cube complex (Chepoi 2000) .
.
.
.
.
.
8 / 27
. Median graph
.
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median.
. Theorem (Chepoi 1996) . .If Γ is a median graph, then 0-Ext[Γ ] is in P.
.
.
.
.
.
.
8 / 27
. Metric relaxation (Karzanov 1998)
. 0-Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
.
s.t.
(X , d): 0-extension of (VΓ , dΓ )
.
.
.
.
.
.
9 / 27
. Metric relaxation (Karzanov 1998)
. Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
.
s.t.
(X , d): extension of (VΓ , dΓ )
.
.
.
.
.
.
9 / 27
. Metric relaxation (Karzanov 1998) . Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
s.t.
d(x, x) = 0
(x ∈ X )
d(x, y ) = d(y , x) ≥ 0 (x, y ∈ X ) d(x, y ) + d(y , z) ≥ d(x, z) (x, y , z ∈ X ) .
d(s, t) = dΓ (s, t)
(s, t ∈ VΓ ⊆ X )
Rem: Ext[Γ ] is polynomial size LP → P
.
.
.
.
.
.
9 / 27
. Metric relaxation (Karzanov 1998) . Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
s.t.
d(x, x) = 0
(x ∈ X )
d(x, y ) = d(y , x) ≥ 0 (x, y ∈ X ) d(x, y ) + d(y , z) ≥ d(x, z) (x, y , z ∈ X ) .
d(s, t) = dΓ (s, t)
(s, t ∈ VΓ ⊆ X )
Rem: Ext[Γ ] is polynomial size LP → P Q. What is Γ for which Ext[Γ ] is exact (for every X , c) ?
c.f. Multicommodity flows (Karzanov 1998, H. 2009 ∼) .
.
.
.
.
.
9 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle .
Rem: frame is obtained by gluing K2,m and K. 2 (in. a certain way) . . .
.
10 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle .
Rem: frame is obtained by gluing K2,m and K (in. a certain way) . 2 . . .
.
10 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle . c.f. frame ≃ CAT(0)-complex of folders (Chepoi 2000)
.
.
.
.
.
.
10 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle . . Theorem (Karzanov 1998) . Γ . is a frame if and only if Ext[Γ ] = 0-Ext[Γ ]. . Corollary (Karzanov 1998) . .If Γ is a frame, then 0-Ext[Γ ] is in P.
.
.
.
.
.
.
10 / 27
Rem. {frames} is not closed under Cartesian product Rem. frame ̸= median graph ∈ {median graphs}, ̸∈ {frames} ̸∈ {median graphs}, ∈ {frames}
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.
.
.
.
.
11 / 27
Rem. {frames} is not closed under Cartesian product Rem. frame ̸= median graph ∈ {median graphs}, ̸∈ {frames} ̸∈ {median graphs}, ∈ {frames} .
Γ : modular ⇔ ∀ triple has a median. Γ : orientable ⇔ ∃ orientation: ∀ 4-cycle
. Rem. frame = orientable hereditary modular graph (c.f. Bandelt 85) Ex. Hasse diagram of modular lattice
.
.
.
.
.
.
11 / 27
. Hardness result . Theorem (Karzanov 1998) . If . Γ is not modular or not orientable, then 0-Ext[Γ ] is NP-hard.
NP-hard orientable modular
? median graph
frame
P
P
.
.
.
.
.
.
12 / 27
. Partial result & conjectures (Karzanov 2004)
NP-hard orientable modular NP-hard?
P
P?
median graph
frame
P
P
.
.
.
.
.
.
13 / 27
. Main result . Theorem (H. 2012, SODA 2013) . If . Γ is orientable modular, then 0-Ext[Γ ] is in P.
NP-hard orientable modular
.
.
.
.
.
.
14 / 27
. Proof idea: Discrete Convex Analysis & Valued-CSP
Discrete Convex Analysis (Murota 1996 ∼) → Our approach suggests “Discrete Convex Analysis on Γ ”
.
.
.
.
.
.
15 / 27
. Proof idea: Discrete Convex Analysis & Valued-CSP
Discrete Convex Analysis (Murota 1996 ∼) → Our approach suggests “Discrete Convex Analysis on Γ ” Valued-CSP ≃ Minimization of a sum of functions with bounded arity ∑ Min. f i1 ,i2 ,...,iK (ρi1 , ρi2 , . . . , ρiK ) 1≤i1
.
Discrete Convexity and Polynomial Solvability in Minimum 0-Extension Problems Hiroshi HIRAI The University of Tokyo [email protected]
RIMS, Kyoto, October 17, 2012
.
.
.
.
.
.
1 / 27
. Contents Minimum 0-extension problems Known results on P/NP-hard classification Main result Proof idea from Discrete Convex Analysis & Valued-CSP Some technical detail . Notation . (semi)metric d on X ⇔ d : X × X → R+ , d(x, x) = 0, .
d(x, y ) = d(y , x),
d(x, z)+d(z, y ) ≥ d(x, y ).
(X , d): (semi)metric space
.
.
.
.
.
.
2 / 27
. Minimum 0-Extension Problem (Karzanov 1998) (S, µ): finite metric space
Def: extension of (S, µ) ⇔ metric space (X , d) s.t. X ⊇ S and d|S = µ Def: 0-extension of (S, µ) ⇔ extension (X , d) s.t. ∃ρ : X → S: ρ|S = id and d = µ ◦ ρ (X, d)
(X, d)
µ(u, s)
µ(s, t)
s u
t (S, µ)
ρ µ(u, t)
extension
0-extension
.
.
.
.
.
.
3 / 27
. Minimum 0-Extension Problem (Karzanov 1998) (S, µ): finite metric space Def: extension of (S, µ) ⇔ metric space (X , d) s.t. X ⊇ S and d|S = µ Def: 0-extension of (S, µ) ⇔ extension (X , d) s.t. ∃ρ : X → S: ρ|S = id and d = µ ◦ ρ . Minimum 0-Extension Problem on (S, µ) . ( ) Given a finite set X ⊇ S and c : X2 → Q+ , minimize
∑
c(xy )d(x, y )
xy
.
subject to
(X , d): 0-extension of (S, µ)
.
.
.
.
.
.
3 / 27
. 0-Ext[µ]
Minimize
∑
c(xy )µ(ρ(x), ρ(y ))
xy
.
subject to
ρ : X → S, ρ|S = id.
.
.
.
.
.
.
4 / 27
. 0-Ext[µ]
Minimize
∑
c(xy )µ(ρ(x), ρ(y ))
xy
subject to
.
ρ : X → S, ρ|S = id.
. → MRF optimization form: ∑∑ ∑ Minimize ciq µ(q, ρi ) + cij µ(ρi , ρj ) i
.
subject to
q∈S
i<j
(ρ1 , ρ2 , . . . , ρn ) ∈ S × S × · · · × S
Applications: computer vision, clustering, learning theory,...
.
.
.
.
.
.
4 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
.
subject to
ρ : X → S, ρ|S = id.
.
.
.
.
.
.
5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
.
subject to
ρ : X → S, ρ|S = id.
s
Γ = K2
t
X
.
.
.
.
.
.
5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ c(xy )dΓ (ρ(x), ρ(y )) 0-Ext[Γ ] Minimize xy
.
subject to
ρ : X → S, ρ|S = id.
Minimum cut s
Γ = K2 1
t
X
.
.
.
.
.
.
5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
.
subject to
ρ : X → S, ρ|S = id.
Γ = K
X .
.
.
.
.
.
5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ c(xy )dΓ (ρ(x), ρ(y )) 0-Ext[Γ ] Minimize xy
.
subject to
ρ : X → S, ρ|S = id.
Multi-terminal cut
Γ = K
1
X .
.
.
.
.
.
5 / 27
(S, µ) := (VΓ , dΓ ) for undirected graph Γ . ∑ 0-Ext[Γ ] Minimize c(xy )dΓ (ρ(x), ρ(y )) xy
subject to
.
ρ : X → S, ρ|S = id.
Γ 2 1
X .
.
.
.
.
.
5 / 27
. Tractability Question (Karzanov 1998, 2004)
Γ = K2 ⇒ minimum cut ⇒ P Γ = Kn ⇒ multi-terminal cut ⇒ NP-hard . Question . What is Γ for which 0-Ext[Γ ] is in P ? .
(n ≥ 3)
.
.
.
.
.
.
6 / 27
. A classical result in location theory . Theorem (Picard-Ratliff 1978) . If . Γ is a tree, then 0-Ext[Γ ] is in P.
.
.
.
.
.
.
7 / 27
. A classical result in location theory . Theorem (Picard-Ratliff 1978) . If . Γ is a tree, then 0-Ext[Γ ] is in P. Obs. Γ ∈ P and Γ ′ ∈ P ⇒ Γ × Γ ′ ∈ P (× := Cartesian product)
Γ
Γ′
Γ × Γ′ .
.
.
.
.
.
7 / 27
. Median graph .
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median. u
y x
y
x z w z
v
.
.
.
.
.
.
8 / 27
. Median graph .
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median.
c.f. Median graph ≃ graph of CAT(0) cube complex (Chepoi 2000) .
.
.
.
.
.
8 / 27
. Median graph .
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median.
median
c.f. Median graph ≃ graph of CAT(0) cube complex (Chepoi 2000) .
.
.
.
.
.
8 / 27
. Median graph
.
Median of x1 , x2 , x3 ⇔ v ∈ VΓ satisfying (1 ≤ i < j ≤ 3)
dΓ (xi , xj ) = dΓ (xi , v ) + dΓ (v , xj ) .
Median graph ⇔ ∀ triple has a unique median.
. Theorem (Chepoi 1996) . .If Γ is a median graph, then 0-Ext[Γ ] is in P.
.
.
.
.
.
.
8 / 27
. Metric relaxation (Karzanov 1998)
. 0-Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
.
s.t.
(X , d): 0-extension of (VΓ , dΓ )
.
.
.
.
.
.
9 / 27
. Metric relaxation (Karzanov 1998)
. Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
.
s.t.
(X , d): extension of (VΓ , dΓ )
.
.
.
.
.
.
9 / 27
. Metric relaxation (Karzanov 1998) . Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
s.t.
d(x, x) = 0
(x ∈ X )
d(x, y ) = d(y , x) ≥ 0 (x, y ∈ X ) d(x, y ) + d(y , z) ≥ d(x, z) (x, y , z ∈ X ) .
d(s, t) = dΓ (s, t)
(s, t ∈ VΓ ⊆ X )
Rem: Ext[Γ ] is polynomial size LP → P
.
.
.
.
.
.
9 / 27
. Metric relaxation (Karzanov 1998) . Ext[Γ ] :
Min.
∑
c(xy )d(x, y )
xy
s.t.
d(x, x) = 0
(x ∈ X )
d(x, y ) = d(y , x) ≥ 0 (x, y ∈ X ) d(x, y ) + d(y , z) ≥ d(x, z) (x, y , z ∈ X ) .
d(s, t) = dΓ (s, t)
(s, t ∈ VΓ ⊆ X )
Rem: Ext[Γ ] is polynomial size LP → P Q. What is Γ for which Ext[Γ ] is exact (for every X , c) ?
c.f. Multicommodity flows (Karzanov 1998, H. 2009 ∼) .
.
.
.
.
.
9 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle .
Rem: frame is obtained by gluing K2,m and K. 2 (in. a certain way) . . .
.
10 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle .
Rem: frame is obtained by gluing K2,m and K (in. a certain way) . 2 . . .
.
10 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle . c.f. frame ≃ CAT(0)-complex of folders (Chepoi 2000)
.
.
.
.
.
.
10 / 27
. Frame = graph for which Ext[Γ ] is exact . Γ : frame ⇔ . bipartite no isometric cycle of length > 4 orientable ⇔ ∃ orientation: ∀ 4-cycle . . Theorem (Karzanov 1998) . Γ . is a frame if and only if Ext[Γ ] = 0-Ext[Γ ]. . Corollary (Karzanov 1998) . .If Γ is a frame, then 0-Ext[Γ ] is in P.
.
.
.
.
.
.
10 / 27
Rem. {frames} is not closed under Cartesian product Rem. frame ̸= median graph ∈ {median graphs}, ̸∈ {frames} ̸∈ {median graphs}, ∈ {frames}
.
.
.
.
.
.
11 / 27
Rem. {frames} is not closed under Cartesian product Rem. frame ̸= median graph ∈ {median graphs}, ̸∈ {frames} ̸∈ {median graphs}, ∈ {frames} .
Γ : modular ⇔ ∀ triple has a median. Γ : orientable ⇔ ∃ orientation: ∀ 4-cycle
. Rem. frame = orientable hereditary modular graph (c.f. Bandelt 85) Ex. Hasse diagram of modular lattice
.
.
.
.
.
.
11 / 27
. Hardness result . Theorem (Karzanov 1998) . If . Γ is not modular or not orientable, then 0-Ext[Γ ] is NP-hard.
NP-hard orientable modular
? median graph
frame
P
P
.
.
.
.
.
.
12 / 27
. Partial result & conjectures (Karzanov 2004)
NP-hard orientable modular NP-hard?
P
P?
median graph
frame
P
P
.
.
.
.
.
.
13 / 27
. Main result . Theorem (H. 2012, SODA 2013) . If . Γ is orientable modular, then 0-Ext[Γ ] is in P.
NP-hard orientable modular
.
.
.
.
.
.
14 / 27
. Proof idea: Discrete Convex Analysis & Valued-CSP
Discrete Convex Analysis (Murota 1996 ∼) → Our approach suggests “Discrete Convex Analysis on Γ ”
.
.
.
.
.
.
15 / 27
. Proof idea: Discrete Convex Analysis & Valued-CSP
Discrete Convex Analysis (Murota 1996 ∼) → Our approach suggests “Discrete Convex Analysis on Γ ” Valued-CSP ≃ Minimization of a sum of functions with bounded arity ∑ Min. f i1 ,i2 ,...,iK (ρi1 , ρi2 , . . . , ρiK ) 1≤i1
