On cardinality constrained polymatroids

Discrete Applied Mathematics 164 (2014) 237–245

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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

On cardinality constrained polymatroids Jean François Maurras a , Ingo Spiegelberg b , Rüdiger Stephan c,∗ a

Laboratoire d’Informatique Fondamentale, UMR 6166, Université de la Mediterranée, Faculté des sciences de Luminy, 163 Avenue de Luminy, 13288 Marseille, France b Zuse Institute Berlin, Takustr. 7, D-14195 Berlin, Germany c

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

article

info

Article history: Received 15 October 2010 Received in revised form 29 September 2011 Accepted 8 October 2011 Available online 1 November 2011 Keywords: Polymatroid Cardinality constraints Linear descriptions Facets

abstract This paper extends results on the cardinality constrained matroid polytope presented in Maurras and Stephan (2011) [8] to polymatroids and the intersection of two polymtroids. Given a polymatroid Pf (S ) defined by an integer submodular function f on some set S and an increasing finite sequence c of natural numbers, the cardinality constrained polymatroid is the convex hull of the integer points x ∈ Pf (S ) whose sum of all entries is a member of c. We give a complete linear description for this polytope, characterize some facets of the cardinality constrained version of Pf (S ), and briefly investigate the separation problem for this polytope. Moreover, we extend the results to the intersection of two polymatroids. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Given a combinatorial optimization problem Π and an increasing finite sequence c of nonnegative integer numbers, we obtain a cardinality constrained version Πc of Π by permitting only those feasible solutions of Π whose cardinalities, that is the number of their elements, are members of c. Maurras [7] and Camion and Maurras [1] started in the 1980’s with the polyhedral investigation of those problems. They introduced a family of inequalities, called forbidden cardinality inequalities, to cut off solutions of forbidden cardinality. Let E be a finite set, and let I ⊆ 2E be the set of feasible solutions of the combinatorial optimization problem. Then, this family consists of the inequalities

(cp+1 − cp )x(F ) − (|F | − cp )x(E ) ≤ cp (cp+1 − |F |) for all F ⊆ E with cp < |F | < cp+1 for some p ∈ {1, 2, . . . , m − 1}.

(1)

Here, c := (c1 , c2 , . . . , cm ), and for any subset F of E, x(F ) := e∈F xe . These inequalities have been independently rediscovered by Grötschel [4]. Like the trivial inequalities xe ≤ 1 for Π , inequalities (1) are always valid for Πc ; however, they are usually not facet defining for the polyhedron associated with Πc . Nevertheless, the polyhedral analysis of a couple of cardinality constrained combinatorial optimization problems indicates that inequalities (1) often can be used as a template in order to derive strong valid inequalities that incorporate certain combinatorial structures of the given problem Π , see [5,8,10]. For cardinality constrained matroids, this attempt has resulted in a complete linear description for the corresponding polytope [8]. Given a matroid I on E with rank function r : 2E → R, it has been shown that the cardinality constrained matroid polytope, which



∗

Corresponding author. Tel.: +49 30 314 29260. E-mail address: [email protected] (R. Stephan).

0166-218X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2011.10.007

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J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

is the convex hull of the incidence vectors of the sets I ∈ I with |I | = cp for some p, is determined by the following system of linear inequalities: x(F ) ≤ r (F ),

∅ ̸= F ⊆ E ,

(cp+1 − cp )x(F ) − (r (F ) − cp )x(E ) ≤ cp (cp+1 − r (F )), F ⊆ E , cp < r (F ) < cp+1 , p ∈ {1, 2, . . . , m − 1}, c1 ≤ x(E ) ≤ cm , x ≥ 0.

(2)

In this paper, we extend results for cardinality constrained matroids to polymatroids. We provide complete linear descriptions for the cardinality constrained versions of the polymatroid and the intersection of two polymatroids defined on the same ground set. In contrast to [8], where the proof for the complete linear description for the cardinality constrained matroid polytope consists of a quite long list of case by case enumerations, our proofs follow standard techniques as used in [9, Chapters 44,46]. Moreover, we characterize some facets of the polytopes to be considered and close with the investigation of the corresponding separation problems. 2. The cardinality constrained polymatroid A set function f defined on some finite set S, i.e., f : 2S → R, is called submodular if f (T ) + f (U ) ≥ f (T ∩ U ) + f (T ∪ U )

(3)

for all subsets T , U ⊆ S. It is said to be nondecreasing if f (T ) ≤ f (U ) for each T ⊆ U ⊆ S and integer if f (U ) ∈ Z for all U ⊆ S. For instance, the rank function of a matroid is submodular, integer, and nondecreasing. Given a submodular set function f on S, the polytope Pf (S ) := {x ∈ RS : x(U ) ≤ f (U ) for each U ⊆ S , xs ≥ 0 for all s ∈ S } is called the polymatroid associated with f. An increasing finite sequence c = (c1 , c2 , . . . , cm ) of nonnegative integer numbers is called a cardinality sequence. Let f be an integer submodular set function on S. The polytope Pfc (S ) := conv{x ∈ ZS ∩ Pf (S ) : x(S ) = cp for some p ∈ {1, 2, . . . , m}} is said to be the cardinality constrained polymatroid associated with f . The linear program max{w T x : x ∈ Pfc (S )} can be solved in polynomial time with a slight modification of the greedy algorithm for polymatroids. By definition, Pfc (S ) does not contain those vertices x of Pf (S ) such that x(S ) ̸= ci for i = 1, 2, . . . , m. We note, however, that, in difference to cardinality constrained 0/1-polytopes, Pfc (S ) may contain integer points x whose sum of all components is not a member of c. If, for instance, |S | = 2, c = (2, 4), and (2, 0), (0, 4) ∈ Pfc (S ), then 12 (2, 0) + 12 (0, 4) = (1, 2) ∈ Pfc (S ), but 1 + 2 = 3 ̸= ci for i = 1, 2. Let us recall some well known facts on polymatroids. First, Pf (S ) is nonempty if and only if f is nonnegative. Next, for any (not necessarily nondecreasing) nonnegative submodular set function f on S, there exists a unique nondecreasing submodular set function f¯ such that f¯ (∅) = 0 and Pf (S ) = Pf¯ (S ), see, for instance, Schrijver [9, Section 44.4]. Hence, Pfc (S ) is nonempty if and only if f is nonnegative and f¯ (S ) ≥ c1 . 2.1. A complete linear description Let f be a nondecreasing integer submodular set function on S with f (∅) = 0. In this subsection, we show that the cardinality constrained polymatroid Pfc (S ) is determined by the inequalities x(U ) ≤ f (U ) for all U ⊆ S ,

(4)

x ≥ 0,

(5)

c1 ≤ x(S ) ≤ cm ,

(6)

and

(cp+1 − cp )x(U ) − (f (U ) − cp )x(S ) ≤ cp (cp+1 − f (U )) for all U ⊆ S with cp < f (U ) < cp+1

for some p ∈ {1, 2, . . . , m − 1}.

(7)

Inequalities (7) are called f -induced forbidden cardinality inequalities. To prove this result, we first study the single-cardinality case, afterwards the two-cardinality case, and finally the general case.

J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

239

For any q ∈ Z+ , let fq be the q-truncation of f . This means, fq is defined by fq (U ) := min{f (U ), q} for U ⊆ S. We note that (q)

fq is submodular, an integer, and nondecreasing, whenever f is. A complete linear description for Pf (S ) is given by x(U ) ≤ fq (U ), x(S ) = q,

U ( S,

(8)

x ≥ 0,

see [9, Section 44.6e]. This implies that the linear programs (q)

max{w T x : x ∈ Pf (S )}

(9)

and





min yS q +



yU fq (U ) : yU ≥ 0 for U ( S , yS χ S +



U (S

yU χ U ≥ w

(10)

U (S

build a pair of dual LPs. The primal LP (9) can be solved with the greedy algorithm for polymatroids. Sorting the elements s1 , s2 , . . . , sn of S such that w(s1 ) ≥ w(s2 ) ≥ . . . ≥ w(sn ), an optimal solution is xq defined by xqsi := fq (Ui ) − fq (Ui−1 ),

i = 1, 2, . . . , n.

(11)

Here, for any j ∈ {0, 1, . . . , n}, Uj := {s1 , s2 , . . . , sj } denotes the set of the first j elements of S. By [9, Section 44.2], an optimal dual solution is y˜ q defined by q

y˜ Ui := wsi − wsi+1 , q yS q yU

i = 1, 2, . . . , n − 1,

˜ := wsn , ˜ := 0,

otherwise.

Since y˜ is an optimal solution of (10), also yq defined by q

yUi := wsi − wsi+1 , q yS q yU

i = 1, 2, . . . , p(q) − 1,

:= wsp(q) , := 0,

(12)

otherwise, q

is one. Here, p(q) ∈ N such that f (Up(q)−1 ) ≤ q ≤ f (Up(q) ). The vector yq is feasible, since yU ≥ 0 for all U ( S and q

yS χ S +



q

yU χ U = wsp(q) χ S +

p(q)−1



(wsi − wsi+1 )χ Ui

i=1

U (S

n−1 

= wsn χ S +

(wsi − wsi+1 )χ S +

p(q)−1



i=p(q)

≥ wsn χ S +

n −1 

(wsi − wsi+1 )χ Ui

i=1

(wsi − wsi+1 )χ Ui = y˜ qS χ S +

i=1



q

y˜ U χ U ≥ w.

U (S

The optimality of yq follows from the fact that q

yS q +



q

yU fq (U ) = wsp(q) q +

p(q)−1



(wsi − wsi+1 )fq (Ui )

i =1

U (S

= wsn q +

n−1 

(wsi − wsi+1 )q +

i=p(q)

= wsn q +

p(q)−1



(wsi − wsi+1 )fq (Ui )

i =1

n−1   q (wsi − wsi+1 )fq (Ui ) = y˜ qS q + y˜ U fq (U ). i=1

U (S

Next, consider the two-cardinality case. Theorem 2.1. Let f be a nondecreasing integer submodular set function on S with f (∅) = 0, and let k, ℓ be natural numbers (k,e) such that 0 ≤ k < ℓ ≤ f (S ). Then, the cardinality constrained polymatroid Pf (S ) is determined by the inequalities (4), (5), k ≤ x(S ) ≤ ℓ,

(13)

(ℓ − k)x(U ) − (f (U ) − k)x(S ) ≤ k(ℓ − f (U )) for all U ⊆ S with k < f (U ) < ℓ.

(14)

and

(k,ℓ)

Proof. Let P be the polytope given by inequalities (4), (5), (13) and (14). We have to show that Pf

(S ) = P.

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J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

(k,ℓ)

(S ) is a subset of P, one only has to verify that the inequalities determining P are valid for Pf (S ). Inequalities (4), (5) are valid for Pf(k,ℓ) (S ), since they determine Pf (S ), and since Pf(k,ℓ) (S ) is a subset of Pf (S ). Also, (k,ℓ) the cardinality bounds (13) are, of course, valid for Pf (S ). (k,ℓ) It remains to show that inequalities (14) are valid for Pf (S ). Since Pf(k,ℓ) (S ) is the convex hull of the integer points of (k) (ℓ) (k) the subpolytopes Pf (S ) and Pf (S ), it suffices to prove that inequalities (14) are valid for these subpolytopes. For Pf (S ), In order to prove that Pf

(k,ℓ)

this follows from the fact that the inequality associated with U among (14) is equivalent to

(ℓ − f (U ))x(U ) − (f (U ) − k)x(S \ U ) ≤ k(ℓ − f (U )) (k)

and that x(U ) ≤ k, since all x ∈ Pf (S ) satisfy the equation x(S ) = k. Thus, (14) is a conical combination of the valid

inequalities x(U ) ≤ k and −xs ≤ 0, s ∈ S \ U. For Pfℓ (S ), this follows from the fact that the right hand side of this inequality is equal to (ℓ − k)f (U ) − (f (U ) − k)ℓ. Thus, this inequality is the sum of a positive multiple of the inequality x(U ) ≤ f (U ) and a multiple of the equation x(S ) = ℓ. (k,ℓ) Next, we show that P = Pf (S ). First, note that an inequality x(U ) ≤ f (U ), with ∅ ̸= U ( S, is redundant whenever f (U ) ≥ ℓ. Next, for each U with k < f (U ) < ℓ, we have two inequalities, one among (4) and one among (14). The previous arguments used to prove the validity of inequalities (14) imply that the inequalities x(U ) ≤ f (U ) for those U are also redundant. Hence, system (4), (5), (13) and (14) reduces to x(U ) ≤ f (U ), x(U ) −

U ∈ S′,

f (U ) − k

ℓ−k −x(S ) ≤ −k, x(S ) ≤ ℓ, x ≥ 0,

x(S ) ≤ f (U ) − ℓ

f (U ) − k

ℓ−k

,

U ∈ S ′′ , (15)

where S ′ := {U ⊆ S : 0 < f (U ) ≤ k} and S ′′ := {U ⊆ S : k + 1 < f (U ) < ℓ}. Here, the second class of inequalities is obtained by dividing each inequality among (14) by ℓ − k. For each U, with ∅ ̸= U ( S, we have one inequality among (15), for which we introduce a dual variable yU , and we have two inequalities associated with S, namely −x(S ) ≤ −k and x(S ) ≤ ℓ, for which we introduce the dual variables yS ,k and yS ,ℓ , respectively. Thus, max{w T x : xsatisfies (15)}

(16)

and

 

min

yU f (U ) +

U ∈S ′



 yU

f (U ) − ℓ

U ∈S ′′

f (U ) − k

ℓ−k



− yS ,k k + yS ,ℓ ℓ :

   f ( U ) − k yU χ U − yU χ U + y ≥ 0, χ S + (yS ,ℓ − yS ,k )χ S ≥ w ℓ−k U ∈S ′′ U ∈S ′ 



(k,ℓ)

(17) (k,ℓ)

build a pair of dual LPs. Therefore, showing that max {w T x : x ∈ Pf (S )} is equal to (17), proves that Pf (S ) is determined by (15). (k,ℓ) The solutions xq , q ∈ {k, ℓ}, specified in (11), provide an optimal solution x¯ of the linear program max {w T x : x ∈ Pf (S )}.

One simply chooses the better of the solutions xk , xℓ , that is, x¯ := xk if ωk ≥ ωℓ and x¯ := xℓ otherwise. Here, ωq denotes the objective function value of the solution xq for q ∈ {k, ℓ}. The solutions yq , q ∈ {k, ℓ}, given in (12), are not necessarily feasible for (17). However, they can be amalgamated to an optimal solution y¯ of (17) determined by y¯ Ui := wsi − wsi+1 ,

i = 1, 2, . . . , p(ℓ) − 1,

y¯ S ,k := max{−λ, 0}, y¯ S ,ℓ := max{λ, 0}, y¯ U := 0,

otherwise.

ωℓ −ωk ℓ−k

is a specific value that depends on the objective function values ωk and ωℓ and the difference ℓ − k. Recall Here, λ := that p(ℓ) is a natural number such that f (Up(ℓ)−1 ) ≤ ℓ ≤ f (Up(ℓ) ). First, let us note that p(ℓ)−1



i=p(k)

y¯ Ui = wp(k) − wp(ℓ) = ykS − yℓS

J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

241

and

λ−

p(ℓ)−1

p (ℓ)−1  f (Ui ) ωℓ ωk f (Ui ) ℓ yℓUi = − − y ℓ−k ℓ − k ℓ − k i=p(k) ℓ − k Ui i=p(k)   p (ℓ)−1 p (k)−1 p (ℓ)−1 1 ℓ ℓ k k ℓ f (Ui )yUi + ℓyS − f (Ui )yUi − kyS − f (Ui )yUi = ℓ − k i=1 i =1 i=p(k)

=

ℓyℓS − kykS . ℓ−k

Now let us show that y¯ is feasible. As it is easily seen, y¯ is nonnegative. In addition,

  f (U ) − k S ¯yU χ U − ℓ χ + (¯yS ,ℓ − y¯ S ,k )χ S ℓ−k U ∈S ′′ U ∈S ′   p (ℓ)−1 p (k)−1 f (Ui ) − k S y¯ Ui χ Ui − = y¯ Ui χ Ui + χ + λχ S ℓ−k i=1 i=p(k)   p (ℓ)−1 p (ℓ)−1 p (ℓ)−1 ℓ yUi f (Ui ) k ℓ Ui S ℓ = yUi χ + χ yUi + λ − χS ℓ − k ℓ − k i=1 i=p(k) i=p(k) 

y¯ U χ U +

p(ℓ)−1

=





yℓUi χ Ui +

kykS − kyℓS

i=1 p(ℓ)−1

=



ℓ−k

χS +

ℓyℓS − kykS S χ ℓ−k

yℓUi χ Ui + yℓS χ S ≥ w.

i=1

To conclude the proof, it remains to show that y¯ is optimal. If λ ≥ 0, then



y¯ U f (U ) +

 y¯ U

p(k)−1



y¯ Ui f (Ui ) +

p(ℓ)−1



ℓ

yUi fℓ (Ui ) +

i=1 p(ℓ)−1

=



fℓ (Ui )yℓUi +

i=1 p(ℓ)−1

=



p(ℓ)−1



 y¯ Ui



ℓ−k

f (Ui ) − ℓ

kℓ

ℓ−k kℓ

ℓ−k

p(ℓ)−1



i=p(k)

+ y¯ S ,ℓ ℓ − y¯ S ,k k

f (Ui ) − k

 ℓ

yUi + ℓ λ −

(ykS − yℓS ) −



ℓ−k

i=p(k)

i=1

=

f (U ) − ℓ

f (U ) − k

U ∈S ′′

U ∈S ′

=



p(ℓ)−1



i=p(k)

+ λℓ ℓ

yUi

fℓ (Ui )



ℓ−k

kℓykS − ℓ2 yℓS

ℓ−k

fℓ (Ui )yℓ (Ui ) + ℓyℓS = ωℓ .

i=1

Conversely, if λ < 0, then



y¯ U f (U ) +

U ∈S ′



 y¯ U

f (U ) − ℓ

f (U ) − k

U ∈S ′′ p(k)−1

ℓ−k



+ y¯ S ,ℓ ℓ − y¯ S ,k k

p(ℓ)−1

  (ℓ)−1  y¯ U kℓ p ℓf (Ui ) i = y¯ Ui f (Ui ) + + + λk y¯ U f (Ui ) − ℓ − k i=p(k) i ℓ−k i=1 i=p(k)   p (k)−1 p(ℓ)−1 p (ℓ)−1 ℓ yUi fℓ (Ui ) kℓ  ℓ = ykUi fk (Ui ) + yUi + k λ − ℓ − k ℓ−k i=1 i=p(k) i=p(k) 

p(k)−1

=

 i=1

p(k)−1

=



fk (Ui )ykUi +

kℓykS − kℓyℓS

ℓ−k

−

k2 ykS − kℓyℓS

ℓ−k

fk (Ui )ykUi + kykS = ωk .

i=1

(k,ℓ)

Thus, y¯ is an optimal solution of (17), and therefore, (15) is a complete linear description of Pf

(S ).



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J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

We now formulate our main result. Corollary 2.2. Let f be a nondecreasing integer submodular set function on S with f (∅) = 0, and let c = (c1 , c2 , . . . , cm ) be a cardinality sequence with cm ≤ f (S ). Then, the cardinality constrained polymatroid Pfc (S ) is determined by the inequalities (4)–(7). Proof. Let P be the polytope constituted by (4)–(7). We have to show that Pfc (S ) = P. Pfc (S ) is a subset of P, since all inequalities determining P are valid for Pfc (S ). For the f -induced forbidden cardinality inequalities (7), this can be shown as in [8]. Let U ⋆ be any subset of S such that cp < f (U ⋆ ) < cp+1 for some p ∈ {1, 2, , . . . , m − 1}, and let x⋆ ∈ Pfc (S ) be any integer point with x⋆ (S ) = cq for some q ∈ {1, 2, . . . , m}. If q ≤ p, that is cq ≤ cp , then

(cp+1 − cp )x⋆ (U ⋆ ) − (f (U ⋆ ) − cp )x⋆ (S ) = (cp+1 − f (U ⋆ ))x⋆ (U ⋆ ) − (f (U ⋆ ) − cp )x⋆ (S \ U ⋆ ) ≤ (cp+1 − f (U ⋆ ))x⋆ (U ⋆ ) ≤ cp (cp+1 − f (U ⋆ )), since x⋆ (U ⋆ ) ≤ x⋆ (S ) = cq ≤ cp . Next, if q ≥ p, that is, cq ≥ cp+1 , then

(cp+1 − cp )x⋆ (U ⋆ ) − (f (U ⋆ ) − cp )x⋆ (S ) ≤ (cp+1 − cp )f (U ⋆ ) − (f (U ⋆ ) − cp )cq ≤ (cp+1 − cp )f (U ⋆ ) − (f (U ⋆ ) − cp )cp+1 = cp (cp+1 − f (U ⋆ )). Thus, x⋆ satisfies the inequality among (7) associated with U ⋆ , and hence, Pfc (S ) ⊆ P. Let us show the converse. In the case m = 1 being trivial, we assume that m ≥ 2. This case can be traced back to the twocardinality case as follows. For every x ∈ RS , if x(S ) is in the interval [c1 , cm ], then x(S ) is in one of the subintervals [cp , cp+1 ], p = 1, 2, . . . , m − 1. Thus, x ∈ P particularly implies that x(S ) ∈ [cp , cp+1 ] for some p, and hence, x satisfies inequalities (4) (cp ,cp+1 )

and (5) as well as (13) and (14) for k := cp and ℓ := cp+1 . Thus, x ∈ Pf (cp ,cp+1 )

since Pf

(S ) is a subset of Pfc (S ).

(S ) by Theorem 2.1, and therefore, x ∈ Pfc (S ),



2.2. Facets In this subsection, we briefly sketch the characterization of facets of Pfc (S ). Since the proofs are analogous to that for the cardinality constrained matroid polytope and most of them are rather technical, we skip them and refer the interested reader to [8]. Let f be a submodular set function on S. A subset U ⊆ S is said to be a flat if f (U ∪ {s}) > f (U ) for all s ∈ S \ U, and U is called inseparable if there is no partition of U into nonempty sets U1 and U2 with f (U ) = f (U1 ) + f (U2 ). For any U ⊆ S and any k ∈ {0, 1 . . . , f (S )}, define the number f k (U ) := k − f (S \ U ). Due to the submodularity of f , we have f k (U1 ) + f k (U2 ) ≤ f k (U ) for all partitions {U1 , U2 } of U, and U is said to be k-separable if the equality holds for some U1 ̸= ∅ ̸= U2 , otherwise k-inseparable. Let us first characterize the dimension of Pfc (S ). Recall that, for any nondecreasing integer submodular set function f on S with f (∅) = 0, the polymatroid Pf (S ) is fulldimensional if and only if f ({s}) > 0 for all s ∈ S. For the cardinality constrained polymatroid, we have the following result. Theorem 2.3. Let f be a nondecreasing integer submodular set function on S such that f (U ) = 0 if and only if U = ∅. (k)

(i) For any k ∈ N with 0 < k ≤ f (S ), dim Pf (S ) = |S | − 1 if and only if S is inseparable or k < f (S ). (ii) For any cardinality sequence c = (c1 , c2 , . . . , cm ) with m ≥ 2, Pfc (S ) is fulldimensional unless c = (0, f (S )) and S is separable.  It is well known that, whenever Pf (S ) is fulldimensional, an inequality x(U ) ≤ f (U ) defines a facet of Pf (S ) if and only if U is an f -flat and U is inseparable, see [2] or [9, Section 44.6c]. For Pfc (S ), we have: Theorem 2.4. Let f be a nondecreasing integer submodular set function on S such that f (U ) = 0 if and only if U = ∅. For any ∅ ̸= U ⊆ S, the inequality x(U ) ≤ f (U ) defines a facet of Pfc (S ) if and only if one of the following conditions holds. (i) (ii) (iii) (iv) (v)

0 < f (U ) < cm−1 and U is a flat and inseparable. 0 < cm−1 = f (U ) < cm < f (S ), and U is a flat and inseparable. 0 < cm−1 = f (U ) < cm = f (S ), U is a flat and inseparable, S \ U is cm -inseparable, and S is inseparable. 0 < cm−1 < cm = f (U ), U = S, and cm < f (S ) or S inseparable. cm−1 = c1 = 0, cm = f (S ), and f (U ) + f (S \ U ) = f (S ). 

J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

243

Fig. 1. Directed graph D = (V , A).

Theorem 2.5. Let f be a nondecreasing integer submodular set function on S such that f (U ) = 0 if and only if U = ∅. Moreover, let U ⊆ S with cp < f (U ) < cp+1 for some p ∈ {1, . . . , m − 1}. Then, the f -induced forbidden cardinality inequality associated with U defines a facet of Pfc (S ) if and only if (cp+1 )

(a) cp = c1 = 0 and the inequality x(U ) ≤ f (U ) defines a facet of Pf (S ), or (b) cp > 0, U is closed and (i) cp+1 < f (S ) or (ii) the set S \ U is cp+1 -inseparable.



2.3. Separation Given any x⋆ ∈ RS , the separation problem consists of finding a valid inequality for Pfc (S ) violated by x⋆ if there is any. By default, we may assume that x⋆ satisfies the inequalities x(S ) ≥ c1 , x(S ) ≤ cm , and xs ≥ 0 for all s ∈ S. A violated inequality among the family (4) x(U ) ≤ f (U ) for all ∅ ̸= U ⊆ S can be found with any algorithm minimizing a submodular function. Since f is submodular, also the function g defined by g (U ) := f (U ) − x⋆ (U ) is submodular, and a set U minimizing g provides a violated inequality among (4) if and only if g (U ) < 0, see, for instance, Korte and Vygen [6]. There are several algorithms available that compute the minimum value of a submodular function, given by a value giving oracle, in strongly polynomial time. For surveys, see [9, Chapter 45] and [6, Chapter 14]. The separation problem for the f -induced forbidden cardinality inequalities (7) can be traced back to that for the inequalities (4) in strongly polynomial time. The construction is analogous to that in [8, Section 2.3]. The main observation is that, provided x⋆ satisfies all inequalities (4) and cp < x⋆ (S ) < cp+1 for some p, the f -induced forbidden cardinality inequality associated with U is violated by x⋆ if and only if f (U ) − x′ (U ) < cp

(δ−1) , δ

where δ :=

x⋆ (S )−cp cp+1 −cp

and x′ := 1δ x⋆ .

Theorem 2.6. Let f : S → R be an integer nondecreasing submodular function with f (∅) = 0 given by a value giving oracle. Moreover, let c be a cardinality sequence and x⋆ ∈ RS . Then, the separation problem for x⋆ and Pfc (S ) can be solved in strongly polynomial time.  3. Extensions It stands to reason to investigate the intersection of two cardinality constrained polymatroids dened on the same set S. First, we want to address this task in terms of matroids. By Edmonds [2], if an independence system I dened on some ground set E can be described as the intersection of two matroids I1 and I2 , then the polytope constituted by I is the intersection of the two corresponding matroid polytopes, that is,

{convχ I : I ∈ I} = Pr1 (E ) ∩ Pr2 (E ), where rj denotes the rank function of the matroid Ij for j = 1, 2. In [8], the question has been raised whether or not this is also true for the cardinality constrained version of this equation. The answer is no. To this end, we consider a cardinality constrained version of the branching polytope defined on the directed graph shown in Fig. 1. Given a digraph D = (V , A), a branching is a forest that contains, for every node v ∈ V , at most one arc entering v . Denote c by PBranch (A) the branching polytope, that is the convex hull of the incidence vectors of all branchings of D, and by PBranch (A) its cardinality constrained version. It is well known that the collection of all branchings of D is the intersection of the graphic matroid I1 := {B ⊆ A : B is a forest} and the partition matroid I2 := {B ⊆ A : |B ∩δ − (v)| ≤ 1 ∀v ∈ V }. Here, δ − (v) denotes the set of arcs entering v . Thus, denoting by rj the rank function associated with the matroid Ij for j = 1, 2, Edmonds’ result says that PBranch (A) = Pr1 (A) ∩ Pr2 (A). Let D = (V , A) be now the digraph shown in Fig. 1. Moreover, let x ∈ RA be defined by xa := 21 for all a ∈ A. Then, it (2,6)

(A) and x ∈ Pr(22,6) (A), which implies x ∈ Pr(12,6) (A) ∩ Pr(22,6) (A). However, one also verifies that x ̸∈ PBranch (A), since PBranch (A) = ∅. Given two nonnegative nondecreasing integer submodular set functions f1 , f2 on S with f1 (∅) = f2 (∅) = 0, and given a cardinality sequence c = (c1 , c2 , . . . , cm ), we define the cardinality constrained intersection polytope by is easily seen that x ∈ Pr1 (2,6)

(6)

Pfc1 ∩f2 (S ) := conv{x ∈ ZS ∩ Pf1 (S ) ∩ Pf2 (S ) : x(S ) = cp for some p}.

244

J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

In case that c = (0, 1, . . . , |S |), we write Pf1 ∩f2 (S ) instead of Pfc1 ∩f2 (S ). As matroid polytopes are polymatroids, the above example shows that Pfc1 ∩f2 (S ) ̸= Pfc1 (S ) ∩ Pfc2 (S ), in general. This means that the inequalities describing the cardinality constrained polymatroids Pfc1 (S ) and Pfc2 (S ) are not sufficient to provide a complete linear description of Pfc1 ∩f2 (S ). Equality has been proven only for Pf1 ∩f2 (S ) (see [2]) and the case that c = (k, k + 1, k + 2, . . . , ℓ) (see [9, Section 46.2]), and therefore, equality holds in particular if c = (k). Based on some examples with convex hull codes and the well-known fact that Pf1 ∩f2 (S ) = {x ∈ RS+ : x(U ) ≤ f (U )}, where f (U ) := min(f1 (T ) + f2 (U \ T )), T ⊆U

U ⊆S

(18)

we conjecture that Pfc1 ∩f2 (S ) is determined by the inequalities x(U ) ≤ f (U ),

U ⊆ S,

(cp+1 − cp )x(U ) − (f (U ) − cp )x(S ) ≤ cp (cp+1 − f (U )), p ∈ {1, 2, . . . , m − 1}, c1 ≤ x(S ) ≤ cm , x ≥ 0.

(19)

U ⊆ S , cp < f (U ) < cp+1 ,

If our conjecture is true, then the case m ≥ 2 reduces to the case m = 2 using the same argumentation as in the proof to Corollary 2.2. Unfortunately we do not completely understand the case m = 2. Going back to the cardinality constrained (k,ℓ) (S ), we see that the proof of Theorem 2.1 works, since there is a common optimal dual solution to the linear polymatroid Pf (k)

(ℓ)

programs max w T x s.t. x ∈ Pf (S ) and max w T x s.t. x ∈ Pf (S ). We are not able to extend this result to the intersection of two cardinality constrained polymatroids. In the Appendix we give some results on the structure of optimal dual solutions (k) (k,k+1) to the linear programs max w T x s.t. x ∈ Pf (S ) and max w T x s.t. x ∈ Pf (S ) that may be helpful to prove our conjecture. Appendix We adapt two results from the literature that give some insights into the structure of special dual optimal solutions to (k) (k,k+1) the linear programs max w T x s.t. x ∈ Pf (S ) and max w T x s.t. x ∈ Pf (S ). Our proofs are based on arguments used in [3] and [9, Section 46.7]. q q q For i = 1, 2, let fi be the q-truncation of fi . We recall that fi is defined by fi (U ) := min{fi (U ), q} for all U ⊆ S. One easily q q checks that fi (∅) = 0 and fi is nonnegative (integer, submodular), whenever fi (∅) = 0 and fi is nonnegative (integer, submodular). Defining f q by q

q

f q (U ) := min(f1 (T ) + f2 (U \ T )), T ⊆U

U ⊆ S,

(A.1)

we see that f q (U ) = min{f (U ), q} for all U ⊆ S. Consider the pair of dual linear programs max{w T x : x(U ) ≤ f q (U ) for all U ( S, x(S ) = q, x ≥ 0}

(A.2)

and

 min yS q +

 

yU f (U ) : yU ≥ 0 for all U ( S, yS χ + q

S



U (S

yU χ

U

≥ w.

(A.3)

U (S

q

Lemma A.1. For i = 1, 2, let fi be the q-truncation of a nonnegative integer submodular set function fi on S with fi = ∅, and let the set function f q be defined by (A.1). Then, there exists an optimal solution y of (A.3) of the following form. Let U = {U1 , U2 , . . . , Un } be the collection of subsets U of S with yU > 0. For each Uj ∈ U, there exists a partition {Vj , Wj } of q q Uj with f q (Uj ) = f1 (Vj ) + f2 (Wj ) such that V1 ⊆ V2 ⊆ . . . ⊆ Vn and W1 ⊇ W2 ⊇ . . . ⊇ Wn . Proof. Let R be the collection of all pairs (V , W ) of subsets of S with V ∩ W = ∅. By definition of f q , (A.2) is equivalent to the linear program q

q

max{w T x : x(S ) = q, x ≥ 0, x(V ) + x(W ) ≤ f1 (V ) + f2 (W ) ∀ (V , W ) ∈ R},

(A.4)

J.F. Maurras et al. / Discrete Applied Mathematics 164 (2014) 237–245

245

and hence, (A.3) is equivalent to



 min zS q +



q zV ,W f1

( (V ) +

q f2

(W )) : zV ,W ≥ 0 for all (V , W ) ∈ R, zS χ + S

(V ,W )∈R



z V ,W χ

V ∪W

≥w .

(A.5)

(V ,W )∈R

Let z attain the minimum (A.5) such that



zV ,W (|V | + |S \ W |) (|W | + |S \ V |)

(V ,W )∈R

is minimized. Then, along the lines of the proof to Theorem 46.4 in [9] one verifies that either A ⊆ C and B ⊇ D, or A ⊇ C and B ⊆ D whenever zA,B > 0 and zC ,D > 0. This implies

R0 := {(V , W ) ∈ R : zV ,W > 0} = {(V1 , W1 ), (V2 , W2 ), . . . , (Vn , Wn )} q

q

with V1 ⊆ · · · ⊆ Vn and W1 ⊇ · · · ⊇ Wn . Since z is optimal, it follows that f1 (Vj ) + f2 (Wj ) = f q (Vj ∪ Wj ) for j = 1, 2, . . . , n. Moreover, w.l.o.g. we may assume that the pairs (Vj , Wj ) are pairwise disjoint and that V ∪ W ( S for all (V , W ) ∈ R0 . (Otherwise, there is such a set R0 with smaller n.) Therefore, by setting yU := zV ,W for all U = V ∪ W , with (V , W ) ∈ R0 , yU := zU for U = S, and yU := 0 otherwise, we see that y is as required.  References [1] P. Camion, J.F. Maurras, Polytopes à sommets dans l’ensemble {0, 1}n , Mélanges, hommage á P. Gillis, Cahiers du Centre d’Études de Recherche Opérationnelle, Bruxelles 24 (1982) 107–120. 1982. [2] J. Edmonds, et al., Submodular functions, matroids, and certain polyhedra., in: R. Guy (Ed.), Combinatorial Structures and Their Applications, in: Proc. Calgary Internat. Conf. Combinat. Struct. Appl., Calgary 1969, Gordon and Breach, New York, 1970, pp. 69–87. 1970. [3] H. Groeflin, A.J. Hoffman, On matroid intersections, Combinatorica 1 (1981) 43–47. doi:10.1007/BF02579175. [4] M. Grötschel, Cardinality homogeneous set systems, cycles in matroids, and associated polytopes, in: Grötschel, Martin (Ed.), The sharpest cut. The impact of Manfred Padberg and his work. MPS/SIAM Series on Optimization, 2004, pp. 99–120, 2004. [5] V. Kaibel, R. Stephan, On cardinality constrained cycle and path polytopes, Math. Program. 123 (2(A)) (2010) 371–394. doi:10.1007/s10107-008-02572. [6] B. Korte, J. Vygen, Combinatorial Optimization. Theory and Algorithms, 4th ed., in: Algorithms and Combinatorics, vol. 21, Springer, Berlin, 2008, xvii. [7] J.F. Maurras, An example of dual polytopes in the unit hypercube., stud. integer program., proc. workshop Bonn 1975, Ann. Discrete Math. 1 (1977) 391–392. 1977. [8] J.F. Maurras, R. Stephan, On the cardinality constrained matroid polytope, Networks 57 (3) (2011) 140–146. [9] A. Schrijver, Combinatorial Optimization. Polyhedra and Efficiency (3 volumes), in: Algorithms and Combinatorics, vol. 24, Springer, Berlin, 2003. [10] R. Stephan, Cardinality constrained combinatorial optimization: complexity and polyhedra, Discrete Optim. 7 (3) (2010) 99–113.