A compact representation for minimizers of k-submodular functions1footnote 11footnote 1An earlier version of this paper was presented at the 4th International Symposium on Combinatorial Optimization (ISCO 2016), Vietri sul Mare, Italy, May 16–18, 2016 [13].

A compact representation for minimizers of -submodular functions111An earlier version of this paper was presented at the 4th International Symposium on Combinatorial Optimization (ISCO 2016), Vietri sul Mare, Italy, May 16–18, 2016 [13].

Hiroshi Hirai Department of Mathematical Informatics,
Graduate School of Information Science and Technology,
The University of Tokyo, Tokyo, 113-8656, Japan
Email: {hirai,taihei_oki}@mist.i.u-tokyo.ac.jp
Taihei Oki Department of Mathematical Informatics,
Graduate School of Information Science and Technology,
The University of Tokyo, Tokyo, 113-8656, Japan
Email: {hirai,taihei_oki}@mist.i.u-tokyo.ac.jp
Abstract

A -submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a -submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando–Fujishige’s signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from -submodular functions. We give algorithms to construct the elementary PIP of minimizers of a -submodular function for three cases: (i) a minimizing oracle of is available, (ii) is network-representable, and (iii) arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts -submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.

Keywords: -submodular function, Birkhoff representation theorem, poset with inconsistent pairs (PIP), Potts energy function

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1 Introduction

Minimizers of a submodular function form a distributive lattice, and are compactly represented by a poset (partially ordered set) via Birkhoff representation theorem. This fact reveals a useful hierarchical structure of the minimizers, and is applied to the DM-decomposition of matrices and further refined block-triangular decompositions [22].

In this paper, we address such a Birkhoff-type representation for minimizers of a -submodular function. Here -submodular functions, introduced by Huber–Kolmogorov [14], are functions on defined by submodular-type inequalities. This generalization of (bi)submodular functions has recently gained attention for algorithm design and modeling [9, 11, 12, 17, 18].

Our main result is to establish a compact representation for minimizers of a -submodular function. This can be viewed as a generalization of the above poset representation for submodular functions and Ando–Fujishige’s signed poset representation for bisubmodular functions [1]. A feature of our representation is to utilize a poset with inconsistent pairs (PIP) [2, 4, 23], which is a discrete structure having a stronger power of expression than that of a signed poset. Actually a PIP is a poset endowed with an additional binary relation (inconsistency relation), and is viewed as a poset reformulation of 2-CNF. This concept, also known as an event structure, was first introduced by Nielsen–Plotkin–Winskel [23] as a model of concurrency in theoretical computer science, and was independently considered by Barthelemy–Constantin [4] to establish a Birkhoff-type representation theorem for a median semilattice—a semilattice generalization of a distributive lattice. A PIP was recently rediscovered by Ardila–Owen–Sullivant [2] to represent nonpositively-curved cube complexes; the term “PIP” is due to them.

Our results consist of structural and algorithmic ones, summarized as follows:

Structural results.

We show that minimizers of a -submodular function form a median semilattice (Lemma 3). By a Birkhoff-type representation theorem [4] for median semilattices, the minimizer set is represented by a PIP, where minimizers are encoded into special ideals in the PIP, called consistent ideals. PIPs arising from -submodular functions are rather special. We completely characterize such PIPs (Theorem 7), which we call elementary. This representation is actually compact. We show that the size of the elementary PIP for a -submodular function of variables is (Proposition 5).

Algorithmic results.

We present algorithms to construct the elementary PIP of the minimizers of a -submodular function under the following three situations:

  • A minimizing oracle of is given.

  • is network-representable.

  • arises from a Potts energy function.

For (i), we show that the PIP is obtained by calling the minimizing oracle time (Theorem 13). Notice that a polynomial time algorithm to minimize -submodular functions is not known for the value-oracle model but is known for the valued-CSP model [20]. Our result for (i) is applicable to such a case.

For (ii) (and (iii)), we consider a class of efficiently minimizable -submodular functions considered in [18], where a -submodular function in this class is represented by the cut function in a network of vertices and can be minimized by a minimum-cut computation. We show that the PIP is naturally obtained from the residual graph of a maximum flow in the network (Theorems 15 and 16).

For (iii), we deal with a -submodular function obtained from a -label Potts energy function by adding the -label (meaning “non-labeled”). Such a -submodular function, called Potts -submodular, is particularly useful in vision applications. Indeed, via the persistency property [9, 18], a minimizer of (an optimal labeling) is partly recovered from a minimizer of the relaxation . Gridchyn–Kolmogorov [9] showed that a minimizer of a Potts -submodular function can be obtained by calls of a max-flow algorithm performed on a network of vertices. We show that the PIP is also obtained in the same time complexity (Theorem 17). In showing this result, we reveal an intriguing structure of the PIP for a Potts -submodular function (Theorem 23), and utilize results [10, 15] from undirected multiflow theory.

We also discuss enumeration aspects for minimizers. Maximal minimizers, which are minimizers with a maximum number of nonzero components, are of particular interest from the view of partial optimal labeling. For a Potts -submodular function, we show that the problem of enumerating all maximal minimizers reduces to the problem of enumerating all ideals of a single poset (Theorem 26). This enables us to use an existing fast enumeration algorithm, and leads to a practical algorithm enumerating all maximal partial optimal labeling in actual computer vision problems. We present experimental results for real instances of stereo matching problems.

Organization.

The rest of this paper is organized as follows. In Section 2, we give preliminaries including a Birkhoff-type representation theorem between PIPs and median semilattices. In Section 3, we prove the above-mentioned structural results. In Section 4, we prove algorithmic results. Finally, in Section 5, we describe applications and present experimental results.

2 Preliminaries

For a nonnegative integer , we denote by (with ). For a subset of an ordered set, let denote the minimum element in (if it exists). Let be the set of real numbers and . For a function from a set to , a minimizer of is an element that satisfies for all . The set of minimizers of is simply called the minimizer set of . We assume that posets are always finite, and assume the standard notions of lattice theory, such as join and meet .

2.1 -submodular function

Let be a positive integer. Let denote . The partial order on is defined by if and only if for each . Consider the -product of , where the partial order on is defined as the direct product of and is also denoted by . In this way, and its subsets are regarded as posets. For , the support of is the set of indices with nonzero , and is denoted by :

A -submodular function [14] is a function satisfying the following inequalities

(2.1)

for all . Here the binary operation on is given by

(2.2)

for every and . The operation in (2.1) is defined by changing to in (2.2).

Besides its recent introduction, a -submodular function seems to be recognized when Bouchet [5] introduced multimatroids. Indeed, a -submodular function is a direct generalization of the rank function of a multimatroid, and was suggested by Fujishige [8] in 1995 as a multisubmodular function.

It is not known whether -submodular functions for can be minimized in polynomial time on the standard oracle model. However, some special classes of -submodular functions are efficiently minimizable. For example, Kolmogorov–Thapper–Živný [20] showed that a sum of low-arity -submodular functions can be minimized in polynomial time, where the arity of a function is the number of variables. A nonnegative combination of binary basic -submodular functions, introduced by Iwata–Wahlström–Yoshida [18], can be minimized by computing a minimum -cut on a directed network; see Section 4.2.

A nonempty subset of is said to be -closed if it is closed under the operations and . From (2.1), the following obviously holds.

Lemma 1.

The minimizer set of a -submodular function is -closed.

2.2 Median semilattice and PIP

A key tool for providing a compact representation for -closed sets is a correspondence between median semilattices and PIPs, which was established by Barthélemy–Constantin [4]. A recent paper [6] also contains an exposition of this correspondence.

A median semilattice [29] is a meet-semilattice satisfying the following conditions:

  1. Every principal ideal is a distributive lattice.

  2. For all , if and exist, then exists in .

Note that every distributive lattice is a median semilattice. An element of is said to be join-irreducible if it is not minimum and is not represented as a join of other elements. Let denote the set of join-irreducible elements of .

Next we introduce a poset with inconsistent pairs (PIP). A PIP [2, 4, 23] is a poset endowed with an additional symmetric relation satisfying the following conditions:

  1. For all with , there is no with and .

  2. For all , if and , then .

A PIP is also denoted by a triple . The relation is called an inconsistency relation. Each unordered pair of is called inconsistent if . Note that every inconsistent pair of is incomparable. An inconsistent pair of is said to be minimally inconsistent if , and imply and for all . If is minimally inconsistent, the is particularly denoted by . We can easily check the following properties of the minimal inconsistency relation:

  1. For all with , there is no with and .

  2. For all with and , if and , then and .

Actually, PIPs can also be defined as a triple , where is a binary symmetric relation on a poset satisfying the conditions (MIC1) and (MIC2). In this definition, the inconsistency relation on is obtained by

if and only if there exist with , and

for every . Since both definitions of PIP are equivalent, we will use a convenient one.

For a PIP , an ideal of is said to be consistent if it contains no (minimally) inconsistent pair. Let denote the family of consistent ideals of . Regard as a poset with respect to the inclusion order .

(a) violating (IC1)

(b) violating (IC1)

(c) PIP

(d) violating (IC2)

(e) PIP
Figure 1: Examples of PIPs and non-PIP structures. Solid arrows indicate the orders between elements (drawn from higher elements to lowers). Dotted lines and dashed lines indicate the inconsistency relations. In (a), (b) and (d), labeled elements indicate where the violations of (IC1) and (IC2) are. In (c) and (e), the minimal inconsistency relations are drawn by dashed lines.

Figure 1 shows examples of PIPs and non-PIP structures.

The following theorem establishes a one-to-one correspondence between median semilattices and PIPs.

Theorem 2 ([4, Theorem 2.16]).
  1. Let be a median semilattice and a symmetric binary relation on defined by

    if and only if does not exist in

    for every . Then forms a PIP with inconsistency relation . The consistent ideal family is isomorphic to , and an isomorphism is given by for and .

  2. Let be a PIP. The consistent ideal family forms a median semilattice. The PIP obtained in the same way as (1) is isomorphic to .

The latter part of Theorem 2 (2) is implicit in [4], and follows from Theorem 2 (1) and the fact that for PIPs and , if and are isomorphic, then and are also isomorphic [4, p.57].

Remark 1.

A PIP is an alternative expression of a satisfiable Boolean 2-CNF, where consistent ideals correspond to true assignments. Indeed, for a PIP with , consider the following 2-CNF of Boolean variables :

Then an assignment is true if and only if the set of elements with is a consistent ideal. The reverse construction of a PIP from a 2-CNF satisfiable at is also easily verified.

3 -closed set and elementary PIP

The starting point for a compact representation for -closed sets is the following.

Lemma 3.

Every -closed set is a median semilattice.

Proof.

Let be a -closed set. Then is a semilattice since is a semilattice with minimum element , and the operator coincides with on . We show that satisfies the conditions (MS1) and (MS2).

(MS1). Let be the principal ideal of . For all and , is equal to either 0 or . Therefore, for all , the join exists and it holds . Next let be an injection defined by for every . One can easily see that and for every . In other words, is an isomorphism from to . Since any nonempty subset of closed under and is a distributive lattice ordered by inclusion, is also distributive.

(MS2). Let be such that the join of any two of them exists in . Since and are comparable for any , the join exists in , and coincides with . Finally since is closed under , the join belongs to .

Let be a symmetric binary relation on defined by

if and only if does not exist in

for every . Note that for every , if then is equal to . From Theorem 2 (1) and Lemma 3, we obtain the following.

Theorem 4.

Let be a -closed set. Then forms a PIP with inconsistency relation . The consistent ideal family is isomorphic to , and the isomorphism is given by for and .

00002

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(a) a -closed set on

13002

13102

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00312

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(b) the PIP corresponding to (a)
Figure 2: Example of a -closed set and the corresponding PIP. In (a), elements surrounded by double-lined frames are join-irreducible. In (b), non-minimal inconsistency relations are not drawn.

Figure 2 shows an example of a -closed set and the corresponding PIP.

From Theorem 4, it will turn out that the set of join-irreducible elements of every -closed set does not lose any information about the structure of . That is, non-minimum elements in can be obtained as the join of one or more join-irreducible elements of (notice that we cannot obtain the minimum element of in this way). Therefore we call a PIP-representation of . Furthermore, the following proposition, which will be proved in Section 3.1, says that this representation is actually compact.

Proposition 5.

Let be a -closed set on . The number of join-irreducible elements of is at most .

Theorem 4 states that any -closed set can be represented by a PIP. However, not all PIPs correspond to some -closed sets. A natural question then arises: What class of PIPs represents -closed sets? The main result (Theorem 7) of this section answers this question.

Definition 6.

A PIP is called elementary if it satisfies the following conditions:

  1. is the disjoint union of such that every pair of distinct elements is minimally inconsistent if and only if for some .

  2. For any distinct , if and , there is no element with .

  3. For any distinct , if and , either of the following two holds:

    1. Every pair of and is not comparable.

    2. There exist and such that and for all and .

(a) violating (EP1)

(b) elementary

(c) violating (EP2)

(d) elementary
Figure 3: Examples of elementary PIPs and non-elementary PIPs. In all diagrams, the drawn PIPs satisfy the condition (EP0) with . Each element is filled or not filled according to the corresponding part . Non-minimal inconsistency relations are not drawn in each diagram.

Figure 3 shows examples of elementary PIPs and non-elementary PIPs.

Theorem 7.
  1. For every -closed set , the PIP is elementary.

  2. For every elementary PIP , there is a -closed set isomorphic to .

An elementary PIP corresponds to a -closed set on the product of the most “elementary” median semilattice , whereas general PIP can represent an arbitrary median semilattice (by Theorem 2). This is why we use the term “elementary.”

Remark 2.

Consider an elementary PIP with the property that each has the cardinality at most 2. Such a PIP arises from -closed sets on . If we assign a sign to each element so that two nodes in have a different sign, then the PIP is equivalently transformed into a signed poset [26], which is a certain “acyclic and transitive” bidirected graph and is used by Ando–Fujishige [1] for representing -closed sets in . Then ideals in the signed poset correspond to consistent ideals in the original PIP. In the transformation, elements in the signed poset are nonempty members in . Bidirected edges are given according to an appropriate rule; one can guess the rule from the example in Figure 4. (In this figure, we omit redundant edges derived from the transitive closure.) In this way, one can see that the PIP-representation for -closed sets on is equivalent to the one by Ando–Fujishige [1].

(a) an elementary PIP with assigned signs

(b) the singed poset corresponding to (a)
Figure 4: Example of an elementary PIP on and the corresponding signed poset.

The following corollary of Theorem 2 (2) and Theorem 7 (1) will be used in Section 4.

Corollary 8.

Let be a PIP. If is isomorphic to some -closed set, then is elementary.

The remaining part of this section is devoted to proving Theorem 7. To get a motivation behind the properties of elementary PIPs which we prove below, readers may choose to read Algorithm 2 in Section 4.1 first.

3.1 Proof of Theorem 7 (1)

The proof of Theorem 7 (1) is outlined as follows:

  1. First we define the differential of a join-irreducible element as the difference between and the unique lower cover of .

  2. Next we introduce a normalized -closed set, which is a -closed set such that every differential has exactly one nonzero component. We show that every -closed set is isomorphic to some normalized -closed set. This set gives us a natural partition of join-irreducible elements.

  3. Finally we construct an elementary PIP from the partition.

A -closed set is said to be simple if . Any -closed set can be converted to a simple -closed set without any structural change.

Definition 9.

Let be a simple -closed set. For , we say that is a lower cover of , or covers , if and there is no such that . For a join-irreducible element , there uniquely exists covered by . The differential of is defined by if and if , for each .

The uniqueness of a lower cover of a join-irreducible element can be seen from the fact that if has two or more lower covers, then is obtained as the join of these lower covers.

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(c) a normalized -closed set on

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(d) the PIP corresponding to (c)
Figure 5: Examples of a simple -closed set, a normalized set and the corresponding PIPs. In (b) and (d), the differential of each join-irreducible element is written in italics.

Figure 5 (a) and (b) show examples of a simple -closed set and the corresponding PIP, respectively.

We show some properties about differentials. In what follows, we denote the subset by for and . Note that also forms a -closed set if .

Lemma 10.

Let be a simple -closed set. The following hold:

  1. For every and with , is join-irreducible in and holds.

  2. For every and with , it holds .

  3. For every and , there is at most one join-irreducible element such that .

  4. For every , the differential of has at least one nonzero component.

  5. The map is an injection from to .

Proof.

(1). Let and . Suppose to the contrary that . Then there exist such that and . Since , either or is equal to . This contradicts the assumption that and . Hence is join-irreducible. Moreover, from Definition 9, it holds .

(2). Let and such that . Then . Let be the lower cover of and let be the minimum element of . Suppose that . Then it holds since . Hence we obtain , which claims that by Definition 9. This contradicts the assumption. Thus holds.

(3). Suppose that has a join-irreducible element such that . From (2), it holds . This lemma follows from the uniqueness of the minimum element of .

(4). Assume that has a join-irreducible element such that . Let be the lower cover of . Then must hold for each , which contradicts that .

(5). Let such that . Since from (4), there exists such that . Then follows from (3).

Now Proposition 5 is a consequence of Lemma 10.

Proof of Proposition 5.

It suffices to consider the case where is simple. From Lemmas 10 (3) and (4), it holds . Furthermore, since the map is injective, we have . Hence is at most .

A simple -closed set is said to be normalized if it satisfies for all . Examples of a normalized -closed set and the corresponding PIP are shown in Figure 5 (c) and (d), respectively.

Lemma 11.

For any -closed set , there exists a normalized -closed set that is isomorphic to with respect to the relations and .

Proof.

We can suppose that is simple. We first show:

  1. For , it holds that or .

Suppose to the contrary that there exist such that and for distinct . Then it holds from Lemma 10 (3). Hence we have since . However, both and belong to since , thus it holds from Lemma 10 (2). This is a contradiction.

By (1), we can define an equivalence relation over the index set

as follows:

Then each equivalence class can be “contracted” into a single index without any structural change of as follows. Let be the set of equivalence classes. For , let be the set of join-irreducible elements having the differentials of support . Then, by Lemma 10,

  1. For every and , either for all or there uniquely exists such that for all .

Let . Define by if for , and if for . It is easily verified (from Lemma 10) that the map is injective and preserves and . An irreducible element of is the image of an irreducible element of , and, by construction, has the differential of a singleton support. Thus is a normalized -closed set.

Now we are ready to prove Theorem 7 (1).

Proof of Theorem 7 (1).

By Lemma 11, it suffices to consider the case where is normalized. For every , let . From the definition of normalized -closed sets, forms a partition of (note that may be empty). We show that the PIP satisfies the axiom of elementary PIPs with for every .

(EP0, “only if” part). Let be a minimally inconsistent pair. Then there exists such that . We show . Suppose that . From Definition 9, there exists such that and . Now we have and , which contradict the assumption that and are minimally inconsistent. Therefore holds, and we can show in the same way. Thus holds.

(EP1) is an immediate consequence of the following property:

  • Let be distinct. If there exist and such that , then for all , there exists such that ; in particular if .

We show (). Let and such that . Now it holds and . Let . We have since . Thus does not hold. We show that . If not, is equal to , hence belongs to . Therefore it holds since is the minimum element of from Lemma 10 (2). We have a contradiction here. Let . This belongs to from Lemma 10 (1), and it holds .

(EP2). Let be distinct indices such that and . We can assume that (EP2-1) does not hold, i.e., there exist and such that . Consider . By (), there exist such that and . We show . Suppose not. Since and , we can take such that by () with changing the role of and . Now we have , which contradicts . Therefore and are same elements. Consequently, the required element in (EP2-2) is given by . By changing the role of and , we see that is given by .

(EP0, “if” part). Let be distinct with . Now since , there exists a minimally inconsistent pair such that and . From the “only if” part of (EP0), and belong to for some . If , then we have , , and , which contradict (EP2). Hence and it must hold .

3.2 Proof of Theorem 7 (2)

Let be an elementary PIP with partition of condition (EP0). For every , let , where . Let . For a consistent ideal , let be defined by

Now is well-defined since every consistent ideal of has at most one element in each by (EP0). Let . Then and are clearly isomorphic. Therefore the rest of the proof of Theorem 7 (2) is to show that forms a -closed set.

We define binary operations and on as and

for every . Theorem 7 (2) follows immediately from:

Lemma 12.

For every , it hold , and .

Proof.

Let . Since the consistent ideal family of a PIP is closed under the intersection, also forms a consistent ideal of . In addition, we can easily check that holds.

Next we consider . We show that is a consistent ideal. Suppose that is not an ideal of . There exist and such that . Without loss of generality, we assume . Now also contains since is an ideal. Let such that and . We can take since . Thus is greater than 1, and is also greater than 1 since contradicts the condition (EP1) if . From (EP2-2), there exists such that . It holds since . Now we have , , and . This contradicts . Therefore is an ideal. Finally suppose that includes an inconsistent pair . Since is an ideal, it also includes the minimally inconsistent pair with and . From (EP0), and belong to the same part of the partition. This contradicts the fact that , and thus is a consistent ideal. follows from the definitions of on and on .

4 Algorithms

In this section, we study algorithmic aspects of constructing PIP-representations for the minimizer sets of -submodular functions. Let denote the minimizer set of a function . Let denote the time complexity of an algorithm of a maximum flow (and a minimum cut) in a network of vertices and edges. We assume a standard max-flow algorithm, such as preflow-push algorithm, and hence assume that is not less than ; notice that the current fastest one is an algorithm by Orlin [24].

4.1 By a minimizing oracle

We can obtain the PIP-representation for the minimizer set of a -submodular function by using a minimizing oracle -SFM, which returns a minimizer of and its restrictions. Let be the minimum value of . For and , we define a new -submodular function from by

Namely, is a function obtained by fixing the -th variable of to .

Before describing the main part of our algorithm, we present a subroutine GetMinimumMinimizer in Algorithm 1. This subroutine returns the minimum minimizer of a -submodular function. The validity of this subroutine can be checked by the fact that is equal to if and otherwise it holds . This subroutine calls -SFM at most times.

1:A -submodular function
2:The minimum minimizer of
3:function GetMinimumMinimizer()
4:     
5:     for  do
6:         if  then
7:                             
8:     return
Algorithm 1 Obtain the minimum minimizer of a -submodular function

Algorithm 2 shows a procedure to collect all join-irreducible minimizers of a -submodular function. Let be the minimum minimizer of . The function in Algorithm 2 is defined as for every . Since is equal to if and to if , we can regard as a -submodular function obtained by fixing each -th variable of to if . Note that the minimum values of and are the same. The correctness of this algorithm is based on Lemma 10 (1) and (2). Namely, the set of join-irreducible minimizers of coincides with the set

(4.1)

The algorithm collects each join-irreducible minimizer according to (4.1) by calling GetMinimumMinimizer at most times. Consequently, if a minimizing oracle is available, the minimizer set can also be obtained in polynomial time.

Theorem 13.

The PIP-representation for the minimizer set of a -submodular function is obtained by calls of -SFM.

1:A -submodular function
2:The set of all join-irreducible minimizers of
3:function GetJoinIrreducibleMinimizers()
4:     
5:      the function obtained by fixing the -th variable of to for all
6:     
7:     for  do
8:         for  to  do
9:              if  then
10:                                               
11:     return
Algorithm 2 Collect all join-irreducible minimizers of a -submodular function

4.2 Network-representable -submodular functions

Iwata–Wahlström–Yoshida [18] introduced basic -submodular functions, which form a special class of -submodular functions. They showed a reduction of the minimization problem of a nonnegative combination of binary basic -submodular functions to the minimum cut problem on a directed network. We describe their method and present an algorithm to obtain the PIP-representation for the minimizer set.

Let and be positive integers. We consider a directed network with vertex set , edge set and nonnegative edge capacity . Suppose that consists of source , sink and other vertices , where and . Let for . An -cut of is a subset of such that and . We call an -cut legal if for every . There is a natural bijection from to the set of legal -cuts of defined by

See Figure 6.

Figure 6: Legal cut corresponding to . Vertices in each are from left to right.

For an -cut of , let denote the legal -cut obtained by removing vertices in from for every with . The capacity of is defined as sum of capacities of all edges from to . We say that a network represents a function if it satisfies the following conditions:

  1. There exists a constant such that for all .

  2. It holds for all -cuts of .

From (NR1), the minimum value of is equal to the capacity of a minimum -cut of . For every minimum -cut of , is also a minimum -cut since satisfies the condition (NR2). Therefore is a minimizer of , and a minimum -cut can be computed by maximum flow algorithms. Indeed, Iwata–Wahlström–Yoshida [18] showed that nonnegative combinations of basic -submodular functions are representable by such networks; see Iwamasa [16] for further study on this network construction.

Now we shall consider obtaining the PIP-representation for the minimizer set of a -submodular function represented by a network . The minimizer set of is isomorphic to the family of legal minimum -cuts of ordered by inclusion, where the isomorphism is . It is well-known that the family of (not necessarily legal) minimum -cuts forms a distributive lattice. Thus, by Birkhoff representation theorem, the family is efficiently representable by a poset. Picard–Queyranne [25] showed an algorithm to obtain the poset from the residual graph corresponding to a maximum -flow of . We describe their theorem briefly. For an -flow of , the residual graph corresponding to is a directed graph , where

Theorem 14 ([25, Theorem 1]).

Let be a directed network with and the residual graph corresponding to a maximum -flow of . Let be the set of strongly connected components (sccs) of other than the following:

  1. Sccs reachable from .

  2. Sccs reachable to .

Let be a partial order on defined by

if and only if is reachable from on

for every . The ideal family of the poset is isomorphic to the family of minimum -cuts of ordered by inclusion. The isomorphism is given by , where is the set of vertices reachable from .

Our result is the following.

Theorem 15.

Let be a network representing a -submodular function and the residual graph corresponding to a maximum -flow of . Let be the set of sccs of other than the following:

  1. Sccs reachable from .

  2. Sccs reachable to .

  3. Sccs reachable to an scc containing two or more elements in for some .

  4. Sccs reachable to sccs and such that and for some .

A partial order on is defined in the same way as Theorem 14. Let be a symmetric binary relation on defined as

if and only if there are distinct such that

Then forms an elementary PIP with inconsistency relation . The consistent ideal family of is isomorphic to the minimizer set of , where the isomorphism is .

Proof.

First we prove that is a PIP. We can see that satisfies the condition (IC1) since for every with , an scc reachable to and does not belong to according to the above exclusion rule (4). The condition (IC2) is also satisfied from the definition of the relation . Thus forms a PIP.

Next we show for every consistent ideal of . Let be the poset given in Theorem 14. Note that is a subposet of . We show that is an ideal of . Suppose not. Then there exist and such that is reachable from and meets the above exclusion rules (3) or (4). Now since also satisfies the same exclusion rule, does not belong to . This is a contradiction. Hence is an ideal of , and is a minimum -cut (Theorem 14). Moreover, from the exclusion rule (3) and the definition of , we can see that is legal. Therefore is a minimizer of .

Conversely, let be a minimizer of . Since is a minimum -cut, is an ideal of (Theorem 14). Suppose that . Then there exists which meets the exclusion rule (3) or (4). Suppose that meets the rule (3). Then is reachable to an scc such that for some . Now is not reachable to since is not reachable to . Thus meets the rule (1) or belongs to otherwise. In either case it holds . This contradicts the fact that is legal. A similar argument can also be applied in the case where meets the rule (4). Therefore holds, and is an ideal of since is a subposet of . The consistency of is an immediate consequence of the fact that is legal.

Now we have shown that is a bijection from to . In addition, clearly preserves the orders, hence it is an isomorphism. Finally from Corollary 8, is elementary.

Algorithm 3 shows a procedure to obtain from the residual graph . First we can obtain the sccs of in time, where . Additionally, the exclusion rules (1), (2) and (3) can be applied to the sccs in the same time complexity. Hence it is only the exclusion rule (4) that we should carefully take account of. An efficient way is described in Line 4 to 10 in Algorithm 3. For each scc , the algorithm memorizes the set of vertices reachable to . Now since the size of each is at any moment, Algorithm 3 runs in time. Therefore the time complexity for obtaining from is much less than the one for computing from the network . Consequently, we obtain the following theorem:

1:The residual graph corresponding to a maximum -flow
2:The set of sccs of defined in Theorem 15
3:function ApplyExclusionRules()
4:     
5:     Remove all sccs from which meet the exclusion rules (1), (2) or (3)
6:     for  in the reverse topological order of  do
7:         
8:         
9:         for  do
10:              
11:              if  for some  then
12:                  Remove all sccs from which are reachable to , and go to Line 4                             
13:     return
Algorithm 3 Obtain sccs which do not meet the exclusion rules
Theorem 16.

Let be a -submodular function represented by a network with edges. The PIP-representation for the minimizer set of is obtained in time.

4.3 Potts -submodular functions

Here we consider a practically important subclass of network representable -submodular functions, called Potts -submodular functions. Let be a connected undirected graph on vertex set with , where each edge has a positive edge weight . Let be the set of labels. A Potts -submodular function is a -submodular function of the following form:

(4.2)

where is any -submodular function on for each and is a -submodular function on defined by

for each . A Potts -submodular function is naturally associated with a Potts energy function :

(4.3)

where is any function on for each and is defined by if and if .

Finding a labeling of the minimum Potts energy is NP-hard for but particularly important in computer vision applications. Useful information of optimal labelings of the Potts energy can be extracted from a minimizer of a Potts -submodular function with appropriate -submodular functions . Define each by for and . In this case, is a -submodular relaxation of , and an optimal labeling of is a partially recovered from a minimizer of ; see the next section. Another choice of is: for and . Also in this case, a part of an optimal labeling is obtained from a minimizer of , and coincides with Kovtun’s partial labeling [9, 21].

The goal of this section is to develop a fast algorithm to construct the PIP of a Potts -submodular function . Notice that is network-representable with edges [18]. Therefore we can obtain a minimizer as well as the PIP-representation for in time by the network construction in the previous section. However it is hard to apply this algorithm to the vision application with large in [9]. Gridchyn–Kolmogorov [9] developed an -time algorithm to find a minimizer of . The main theorem in this section is a stronger result that the PIP-representation is also obtained in the same time complexity.

Theorem 17.

The PIP-representation for the minimizer set of is obtained in
time.

The rest of this subsection is devoted to proving this theorem. First we construct a network , different from the one in the previous section. For each , decompose as follows. Let be defined by if and otherwise. Choose a minimizer of . Then is represented as

where and for . We remark that is nonnegative by -submodularity, and that implies .

Let us construct . Starting from , define the edge-capacity of each edge by . Next add new vertices , called terminals. For each , if with , add a new edge of capacity . An edge is called a terminal edge. For each with , add a new vertex and a new edge of capacity . A vertex is called the -fringe of . Let . Let be the set of all fringes, the set of all edges incident to fringes, and the set of all terminal edges. Let