Exhaustive Family of Energies Minimizable Exactly by a Graph Cut
Guillaume Charpiat
Pulsar team, INRIA
Sophia-Antipolis, France
Guillaume.Charpiat@inria.fr
Abstract
Graph cuts are widely used in many fields of computer
vision in order to minimize in small polynomial time com-
plexity certain classes of energies. These specific classes
depend on the way chosen to build the graphs represent-
ing the problems to solve. We study here all possible ways
of building graphs and the associated energies minimized,
leading to the exhaustive family of energies minimizable ex-
actly by a graph cut. To do this, we consider the issue of
coding pixel labels as states of the graph, i.e. the choice of
state interpretations. The family obtained comprises many
new classes, in particular energies that do not satisfy the
submodularity condition, including energies that are even
not permuted-submodular.
A generating subfamily is studied in details, in particular
we propose a canonical form to represent Markov random
fields, which proves useful to recognize energies in this sub-
family in linear complexity almost surely, and then to build
the associated graph in quasilinear time. A few experiments
are performed, to illustrate the new possibilities offered.
Introduction
Graph cut has been a very popular optimization tool in
computer vision for a decade, because of their suitability
to many problems defined on images, and because of their
ability to find, in small polynomial time, global optima of
energies which were mistakenly thought to be NP-hard to
minimize before. The graph cut method consists in re-
writing the energy minimization as a search for a minimal
cut in a graph. The new problem obtained is known in graph
theory as min-cut or maximum flow, and there exist effi-
cient algorithms to solve it [5, 1], provided all weights in
the graph are non-negative. The main question that this ar-
ticle addresses is to draw the list of all energies that can be
minimized in such a way.
Other optimization techniques based on graphs exist,
such as minimum ratio weight cycle, loopy belief propa-
gation or spectral clustering. There might even exist other
techniques to estimate minimal cuts on different kinds of
graphs [4]; this would however be out of the scope of this
article, which focuses on graphs considered by usual mini-
mal cut techniques, i.e. with positive weights.
Graph cuts were originally introduced in computer vi-
sion by [9] in 1989, and gained fame one decade later
[18, 2]. Many new applications, performance improve-
ments, wider classes of possible energies, or related opti-
mization techniques then appeared [14, 23, 15, 21, 6, 8, 3].
In particular, the ability to solve certain classes of binary
classification problems exactly with graph cuts was ex-
tended to the approximate resolution of multi-labeling prob-
lems with α-expansions or (α, β)-swaps [13], and even to
exact resolution in the case of convex interaction terms be-
tween variables [11]. Since the class of energies solvable
with graph cuts is regularly extended, an arising question is
to know whether it would be possible to draw an exhaus-
tive, definitive list which could not be broadened anymore.
We will here focus on energies minimizable exactly, since
including approximate techniques would involve their com-
parison and the quantization of their respective errors.
Most works on exact or approximate minimization with
graph cuts require that pairwise interaction terms (restricted
to any pair of labels) are submodular. However this sub-
modularity condition is specific to particular ways of build-
ing graphs where each label is represented by one node,
and we will see that, in the general case, the constraints
are different. Amongst the energies that we are able to
minimize, most are not submodular or not even permuted-
submodular, and this is a major difference with classical
graph construction and related studies [19, 22, 16, 20] about
expressiveness or recognition of permuted problems, which
once again focus on submodular energies or binary ones [7].
The only exceptions to the submodularity rule in the litera-
ture are based on roof duality [17], thanks to a different cor-
respondence between labels and states of the nodes of the
graph, but at the cost of giving only partial solutions to the
problem, in the sense that they do not guarantee to assign a
label to each unknown variable. We will be interested in all
possible ways of coding label choices into graph partitions,
i.e. binary states of nodes. Once such a list is available, an
issue arises, the one of recognizing whether an energy be-
longs to the list and of building the associated graph.
The article is organized as follows : In a first part, we
discuss all possibilities to build positive-weighted graphs to
represent energies to be minimized, and infer the exhaustive
list F of energies minimizable exactly by graph cuts. We
then focus on a generating subfamily F
0
. In the second part,
we propose a canonical form to represent Markov Random
Fields (MRF), in order to study F
0
and express it in a more
practical way. We will see that F
0
includes new kinds of
energies absent from the literature. In the third part we show
how to recognize quickly energies of F
0
(even after pixel-
dependent permutations of labels). Finally, we perform a
few experiments to illustrate the previous sections.
1. Graphs and interpretations
This section aims at drawing the list of all energies that
can be rewritten as the cost of cuts in a graph with positive
weights. Let E(x
1
, x
2
, . . . , x
p
) be a function of p variables
x
i
, to be minimized globally. Let G be a graph, consisting
of q nodes N
j
and of directed edges between pairs of nodes,
with non-negative weights w
jk
. Let us try to build E and
G together so that cuts in the graph G can be interpreted as
choices of values for all variables x
i
.
If we restrict the study to the case of graphs with a fi-
nite number of nodes, then the number of possible cuts of
a graph is finite, so the number of possible states of all
variables (x
i
)
16i6p
has to be finite. Hence each variable
x
i
has to be constrained to a finite set of possible values
X
i
=
L
1
i
, L
2
i
, L
3
i
, ...
which we will name labels. The
number of labels #X
i
may depend on i.
1.1. Graph states
In order to use s-t min cut techniques such as max-flow
[5], we introduce in the graph two special nodes, the sink
and the source of the flow. It is possible to consider several
sources or/and several sinks, but such a problem can be eas-
ily rewritten as a classical graph with only one source and
one sink. The minimal cut obtained can be seen as a parti-
tion of the nodes of the graph : those which are connected to
the source, and those which are connected to the sink. Thus,
considering a cut is equivalent to picking a state s(N
i
) for
each node of the graph amongst the two possibilities men-
tioned. Nodes can then be seen as binary variables, and
graph cuts as states of all these binary variables together.
Finding a minimal cut is then equivalent to finding a state s
of the graph that solves:
min
s : {N
i
}
16i6q
→ {source, sink}
X
i,j
w
ij
δ
s(N
i
) = source
& s(N
j
) = sink
where δ
P
= 1 if P is true, and 0 otherwise, thus summing
over edges cut. Consequently, algorithms finding minimal
cuts [5, 1] can be used to find global minima of any energies
of binary variables of the form
C(b) =
X
i,j
w
ij
b
i
b
j
where b ∈ {0, 1}
q
and where b
i
= 1 − b
i
, provided all
coefficients w
ij
are non-negative. For any pair of binary
variables {b
i
, b
j
}, their interaction cost matrix has conse-
quently to be antidiagonal and positive, which leads to the
well-known submodularity condition with usual graph con-
structions. Note however that this constraint applies to in-
teraction costs between node states, and not to interaction
costs between labels, that we have not defined yet.
1.2. Interpreting graph states as labels
We have to define a way to interpret graph states so that
we can infer from any possible cut of the graph G, i.e. from
any possible state of the binary variables b
j
related to the
nodes, the corresponding choice of labels for the variables
(x
i
)
16i6p
of the energy E. Let us consider one variable
x
i
: its label has to be a function of the states of the nodes
of the graph. This function might depend on a few nodes
only; let us denote by S
i
the set of nodes upon which x
i
depends, it is a subpart of G. This set S
i
of binary variables
can be ordered arbitrarily : (b
1
i
, b
2
i
, . . . , b
q
i
i
), leading to a
numbering J1, 2
q
i
K of all possible states of the subgraph S
i
as integers expressed in base 2. We will name interpretation
any function from this numbering to the set X
i
of labels
available for the energy variable x
i
, i.e. any function
I
i
: J1, 2
q
i
K −→ J1, #X
i
K.
1.2.1 The simple case
Let us first assume that each node of the graph G belongs
to exactly one subgraph S
i
, i.e. that the subgraphs S
i
form
0 0
0 1
1 0
1 1
L
∅
1
L
2
2
2
b
2 2
b
x
2
1 2
3
S
G
1
I
S
1
b
b
b
1
1
b b b
1
1
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
L
L
L
1
1
2
3
1
2 3
11
2
3
1
1
1
∅
L
5
1
1
L
4
x
1
2
S
2
2
b
2
1
b
2
I
Figure 1. The states of subparts S
i
of the graph G can be repre-
sented by a set of binary variables b
j
i
, which are then interpreted
together as labels L
k
i
of the energy variable x
i
via the interpre-
tation function I
i
. Some states of the subgraph S
i
may never be
reached, in that case they do not need to correspond to a label (∅).
2
a partition of G, as in Figure 1. Then, let us be given a
choice of interpretations I
i
for all variables x
i
. For the sake
of readability, we will suppose in a first step that each in-
terpretation I
i
is a one-to-one correspondence between sub-
graph states (b
1
i
, b
2
i
, . . . , b
q
i
i
) and possible labels of variable
x
i
. Then the binary variable b
k
i
can be seen as a function of
x
i
: it is the k
th
digit of I
−1
i
(x
i
). Thus, the energy C(b)
rewrites as a function E(x) of the variables x
i
, by regroup-
ing interactions between binary variables by subgraphs:
E(x) =
X
i
D
i
(x
i
) +
X
i,j
V
i,j
(x
i
, x
j
) (1)
where
D
i
(x
i
) =
q
i
X
k=1
w
sourcek
i
b
k
i
+ w
ksink
i
b
k
i
+
q
i
X
k=1
q
i
X
k
0
= 1
k
0
6= k
w
kk
0
i
b
k
i
b
k
0
i
and
V
i,j
(x
i
, x
j
) =
q
i
X
k=1
q
j
X
l=1
w
kl
i,j
b
k
i
b
l
j
+ w
kl
i,j
b
k
i
b
l
j
where w
kk
0
i
stands for the weight of the edge from the k-
th to the k
0
-th node in subgraph S
i
, and similarly w
kl
i,j
for
edges between different subgraphs. The source and the sink
were distinguished from other nodes.
Definition 1 : Subfamily F
0
. We will denote by F
0
the
family of all energies that can be written in such a way, for
any finite number of variables x
i
, any finite sets of labels
X
i
, any one-to-one interpretations I
i
and any non-negative
coefficients w. Thus F
0
is the set of all energies minimizable
by a cut in a graph, up to the two previous assumptions, that
(S
i
) is a partition of G, and that ∀ i, I
i
is one-to-one.
1.2.2 Sharing nodes; nodes unused in labeling
The first hypothesis made was that the subgraphs S
i
formed
a partition of the graph G. If instead several subgraphs share
a same node, then this node may be divided into as many in-
stances as subgraphs, with infinite links (in both directions)
between the instances, so that each subgraph can have its
own copy of the node without sharing it, while all copies
share a common state. The problem of overlapping sub-
graphs is thus reduced to the previous case. On the other
hand, there may be nodes in G which do not belong to any
subgraph S
i
, i.e. their states are not taken into account in
any interpretation x
i
. The set of all such nodes forms an-
other subgraph S
supp
, which can be seen as a supplemen-
tary variable x
supp
. Then we have a partition of the graph
and can apply the previous approach. We will obtain an
energy of the same form E
1
(x, x
supp
); since we are in-
terested in minimal cuts, one can consider that the quan-
tity minimized is the infimum over x
supp
of this energy :
E
2
(x) = min
x
supp
E
1
(x, x
supp
).
1.2.3 Impossible states; multiply-represented labels
Some states of a subgraph S
i
might be non-reachable be-
cause they correspond to infinite costs; this happens when
infinite weights are set between nodes of the same subgraph
S
i
, or between one of its nodes and the source or the sink.
The interpretation of such states has no consequence since
they are never reached; one can choose to map them to any
possible label of x
i
, to map them to an additional explicit
label ∅ (which will never be chosen), or just to ignore this
state when defining interpretations. On the opposite, several
different states of a subgraph S
i
may be associated with a
same label L
k
i
of the variable x
i
. Then the cost of choosing
this label will be the minimum of the costs of the different
possible interpretations. Let us consider the same graph but
with a one-to-one interpretation, with supplementary labels
so that each state of each subgraph corresponds to exactly
one label, and denote by X(L
k
i
) the set of new labels cor-
responding to the former label L
k
i
. Then we can define :
E
0
(x) = min
y s.t. y
i
∈X(x
i
)
E(y), where E ∈ F
0
. This en-
ergy E
0
satisfies that its global minimum is reached for a
minimal cut on G.
1.2.4 Complete family
Now that we have removed the only two assumptions made
earlier, we obtain :
Proposition 1 : Exhaustive family. The exhaustive fam-
ily F of all energies minimizable by a graph cut is the set
of functions which either belong to the former family F
0
,
either can be written as an infimum of an energy in F
0
with respect to some of its variables, globally (i.e. over all
possible labels of these variables : min
x
supp
) or partially
(i.e. over a subset of the possible labels of these variables :
min
x
i
∈X
r
i
⊂X
i
).
1.3. Known methods as particular cases
To see how standard graph constructions are included in
F, we summarize in Fig. 2 the main different approaches to
global minimization by graph cuts in the literature. We indi-
cate the number of labels, of nodes per subgraph, as well as
the interpretation function chosen and other particularities.
2. Rewriting in a canonical form
We now aim at expressing energies in the families F
or F
0
in a more suitable way than equation (1) and its
constraints on admissible functions D
i
and V
i,j
. For this,
we will first need to study the space of energies related to
Markov random fields.
2.1. Markov random fields
Let F
M
be the set of all energies E of the kind :
E(x) =
X
i
D
i
(x
i
) +
X
i,j
V
i,j
(x
i
, x
j
) (2)
3
Method #X
i
#S
i
I
i
Notes
Binary graph cut 2 ∀i 1
Id : 0 7→ 0
1 7→ 1
Submodularity required
Ishikawa [11] L ∀i L
bijection
node–label
(fixed numbering)
Infinite weights between nodes to maintain order;
Convex interaction term
Roof duality,
a.k.a. QPBO [17]
3 ∀i:
{0, 1,’?’}
2
(0, 1) 7→ 0
(1, 0) 7→ 1
(0, 0), (1, 1) 7→’?’
Partial answer : ‘?’ = unknown label;
Allows supermodularity on a binary problem
Generating family F
0
(this work)
2
n
i
n
i
any bijection
Number of labels = variable-dependent;
Includes non permuted-submodular energies;
Constraints checked in linear time
Figure 2. Most common approaches to global minimization with graph cuts in the literature, and how they can be understood in this
framework. Number of labels per variable, number of nodes per subgraphs are indicated, as well as the interpretation function.
where each variable x
i
belongs to a finite set of possible
labels X
i
, and where D
i
and V
i,j
are any real-valued func-
tions, of one or two of these variables respectively. Such
energies are related to Markov random fields (MRF). Note
that F
0
⊂ F
M
.
Given an ordering of the possible labelings X
i
of each
variable x
i
, the functions D
i
(·) and V
i,j
(·, ·) can be re-
spectively represented as vectors and matrices with real co-
efficients, since they are real-valued functions defined on
a finite ordered set or on a product of two finite ordered
sets. The k
th
component of the vector D
i
is the local cost
D
i
(L
k
i
) of choosing the k
th
possible label L
k
i
for variable
x
i
, and similarly the (k, l)
th
coefficient of matrix V
i,j
is the
interaction cost V
i,j
(L
k
i
, L
l
j
) between the k
th
possible label
for variable x
i
and the l
th
possible label for x
j
.
2.2. Canonical form
In general, there are many different ways to express a la-
beling cost in the form of equation (2), so that two energies
may have different vectors D
i
and matrices V
i,j
while lead-
ing to the same costs for all possible labelings. For example,
the following two Markov chains C
1
and C
2
are equivalent :
E = D
1
(x
1
)+V
12
(x
1
, x
2
)+D
2
(x
2
)+V
23
(x
2
, x
3
) +D
3
(x
3
)
C
1
2
3
4 7
5 0
−4
1
−1 0
2 2
0
0
C
2
6
0
0 0
8 0
−7
3
0 0
1 0
2
3
We will now rewrite any energy in F
M
so that the sums of
elements of each row, each column of each matrix V
i,j
are
0. To do this, let M
S
be the vector space of matrices of the
form M = (r
l
+c
k
)
k,l
. Such matrices are a sum of a matrix
with identical rows r and of a matrix with identical columns
c. We project V
i,j
orthogonally on M
S
to obtain the row
r
i,j
and the column c
i,j
, and we note V
0
i,j
the remaining
orthogonal part, i.e. V
0
i,j
= V
i,j
− (r
i,j
l
+ c
i,j
k
)
k,l
. Let
D
new
i
= D
i
+
X
j
c
i,j
+
X
j
r
j,i
. (3)
The system obtained (D
new
, V
0
) expresses the same costs
as the original one (D, V ). We also subtract the mean val-
ues of each vector D
new
i
; the system (D
0
, V
0
) thus ob-
tained expresses the same energy functional as (D, V ), up
to a constant.
Proposition 2 : Canonical form. Any MRF energy (i.e.,
in F
M
) E =
P
i
D
i
+
P
i,j
V
i,j
can be rewritten, up to
a constant, as
P
i
D
0
i
+
P
i,j
V
0
i,j
where the mean value of
each vector D
0
i
is 0, as well as the mean value of each line,
each column of each matrix V
0
i,j
. This writing is unique and
will be referred as the canonical form of E.
Proof: Existence by construction. Uniqueness : Let A and
B be two writings of the same energy in canonical forms.
They are equal (up to a constant shift) for all labelings; let
us express them as vectors and matrices for a common or-
dering of labels, and write their difference δE with vectors
D
i
= D
A
i
− D
B
i
and matrices V
i,j
= V
A
i,j
− V
B
i,j
. We
want to show that these vectors and matrices are 0. By
hypothesis, the function δE(x) is a constant and does not
depend on x. Let us consider one particular variable x
i
and fix other variables x
j
. The sum of the terms involv-
ing x
i
is D
i
(x
i
) +
P
j
V
i,j
(x
i
, x
j
) and does not depend
on x
i
(since δE is constant). If we sum this quantity over
possible values x
i
∈ X
i
, we obtain #X
i
times the same
value, but also 0 (mean values of vectors or columns are
0) so that D
i
(x
i
) +
P
j
V
i,j
(x
i
, x
j
) = 0 ∀i, x
i
, x
j
. If we
now average this equality over all values x
j
∈ X
j
for all
j 6= i, we find D
i
(x
i
) = 0 (since row means are 0), so
that
P
j
V
i,j
(x
i
, x
j
) = 0 ∀i, x
i
, x
j
. Then ∀i, j, x
i
, the only
term of this constant sum that involves x
j
is V
i,j
(x
i
, x
j
)
and cannot depend on x
j
. Since moreover the (row) mean
of V
i,j
(x
i
, x
j
) over x
j
is 0, we have V
i,j
(x
i
, x
j
) = 0. Con-
sequently, D
A
=D
B
and V
A
=V
B
, hence uniqueness. ut
4
2.3. The energies of the family, in canonical form
Energies of the family F
0
are written in (1) as linear
combinations of quantities b
k
i
, b
k
i
b
k
0
i
and b
k
i
b
l
j
, which can
be seen as vectors or matrices as functions of the subgraph
states representing x
i
and x
j
. Their canonization, detailed
in [24], leads respectively to vectors A
k
i
, A
k,k
0
i
and matri-
ces W
k,l
i,j
defined by (A
k
i
)
s
= (−1)
1+b
k
(s)
, (A
k,k
0
i
)
s
=
(−1)
1+b
k
(s)+b
k
0
(s)
, and W
k,l
i,j
= −A
k
i
⊗ A
l
j
, with b
k
(s)
standing for the k
th
bit of the binary writing of the state s.
For a given i, the vectors A
k
i
and A
k,k
0
i
form an orthogonal
family of q
i
(q
i
+ 1)/2 elements. For given i and j, the ma-
trices W
k,l
i,j
form an orthogonal family too, of q
i
q
j
elements.
We now rewrite F
0
in a more practical way. Let E be
an energy in F
0
as in equation (1) with satisfied constraints,
with a given ordering of the nodes in each subgraph. The
canonical form of E is then :
D
0
i
=
q
i
X
k=1
α
k
i
A
k
i
+
q
i
X
k=1
q
i
X
k
0
= 1
k
0
6= k
α
k,k
0
i
A
k,k
0
i
V
0
i,j
=
q
i
X
k=1
q
j
X
l=1
α
k,l
i,j
W
k,l
i,j
(4)
where
α
k
i
=
1
2
(w
sourcek
i
− w
ksink
i
) +
1
4
X
k
0
(w
kk
0
i
− w
kk
0
i
)
+
1
4
X
j
(w
kk
0
i,j
− w
kk
0
i,j
)
α
k,k
0
i
=
1
4
(w
kk
0
i
+ w
kk
0
i
)
α
k,l
i,j
=
1
4
(w
kl
i,j
+ w
kl
i,j
)
can be seen as parameters of the energy E, i.e. as degrees
of freedom in the design of energies within F
0
.
Note: no degree of freedom is lost by assuming w
kk
0
i
=
w
kk
0
i
and w
kl
i,j
= w
kl
i,j
, thanks to w
sourcek
i
and w
ksink
i
which make the span of α
k
i
to be fully R. Thus it is sufficient
to consider only undirected graphs G .
Proposition 3 : Characterization of F
0
. The family F
0
is the set of MRF energies such that there exist an ordering
of the labels, and parameters α
k
i
∈ R and α
k,k
0
i
, α
k,l
i,j
∈
R
+
, for which the canonical expression (D
0
i
, V
0
i,j
) of the
energy satisfies equations (4).
Let us detail the vectors A
k
, A
k,k
0
and the matrices W
k,l
which generate the span of energies minimizable by a graph
cut. If a subgraph contains only #S
i
= 1 node, then A
1
=
−1
1
and since α
1
∈ R, all canonized potentials D
0
i
of a
binary variable are representable. In the case of #S
i
= 2
nodes :
A
1
i
=
−1
−1
1
1
, A
2
i
=
−1
1
−1
1
, A
1,2
i
=
−1
1
1
−1
and since the dimension of the space of canonized potentials
is 3, with α
1
, α
2
∈ R and α
1,2
∈ R
+
, half of them are
representable for any fixed interpretation. For #S
i
= 3 :
0
B
B
B
B
B
B
B
B
B
@
−1
−1
−1
−1
1
1
1
1
1
C
C
C
C
C
C
C
C
C
A
,
0
B
B
B
B
B
B
B
B
B
@
−1
−1
1
1
−1
−1
1
1
1
C
C
C
C
C
C
C
C
C
A
,
0
B
B
B
B
B
B
B
B
B
@
−1
1
−1
1
−1
1
−1
1
1
C
C
C
C
C
C
C
C
C
A
,
0
B
B
B
B
B
B
B
B
B
@
−1
−1
1
1
1
1
−1
−1
1
C
C
C
C
C
C
C
C
C
A
,
0
B
B
B
B
B
B
B
B
B
@
−1
1
−1
1
1
−1
1
−1
1
C
C
C
C
C
C
C
C
C
A
,
0
B
B
B
B
B
B
B
B
B
@
−1
1
1
−1
−1
1
1
−1
1
C
C
C
C
C
C
C
C
C
A
.
A
1
i
A
2
i
A
3
i
A
1,2
i
A
1,3
i
A
2,3
i
6 degrees of freedom are available (amongst 7). Concerning
interaction matrices : for a pair of subgraphs containing one
node each, there is only one interaction matrix generator :
W
1,1
1,2
=
−1 1
1 −1
.
The space of all canonized interaction matrices M (for any
MRF, not necessarily in F
0
) has only one degree of free-
dom, so that any interaction matrix M writes necessarily as
αW
1,1
for an α ∈ R. If α > 0, M is said to be submodular,
if α 6 0, M is supermodular. Since α
1,1
has to be positive,
any energy of binary variables which is in F
0
has to satisfy
that there exists a label ordering for which V
0
i,j
is submod-
ular for all (i, j) simultaneously. There are however other
interpretations for which given V
0
i,j
are supermodular.
The matrices available in F
0
for the interaction between
a subgraph of 1 node and one of 2 nodes are :
−1 −1 1 1
1 1 −1 −1
,
−1 1 −1 1
1 −1 1 −1
.
W
1,1
1,2
W
1,2
1,2
For interactions between two subgraphs of 2 nodes each :
0
B
@
−1 −1 1 1
−1 −1 1 1
1 1 −1 −1
1 1 −1 −1
1
C
A
,
0
B
@
−1 1 −1 1
−1 1 −1 1
1 −1 1 −1
1 −1 1 −1
1
C
A
,
0
B
@
−1 −1 1 1
1 1 −1 −1
−1 −1 1 1
1 1 −1 −1
1
C
A
,
0
B
@
−1 1 −1 1
1 −1 1 −1
−1 1 −1 1
1 −1 1 −1
1
C
A
.
W
1,1
1,2
W
1,2
1,2
W
2,1
1,2
W
2,2
1,2
Note that in the three last matrices, there exist 2 × 2 sub-
matrices which are supermodular. Moreover, many positive
linear combinations of these matrices are not permuted-
submodular, i.e. there exists no interpretation (= permuta-
tion) that makes all 2 × 2 submatrices become submodular.
Consider e.g. α
1,1
= α
2,2
= 1/2, α
2,1
= α
1,2
= 0 :
5
