Journal ArticleDOI

# The b-branching problem in digraphs

15 Sep 2020-Discrete Applied Mathematics (Elsevier)-Vol. 283, pp 565-576

TL;DR: It is demonstrated that b -branchings yield an appropriate generalization of branchings by extending several classic results on branchings, and a packing theorem extending Edmonds’ disjoint branchings theorem is proved and the integer decomposition property of the b - Branching polytope is proved.

AbstractIn this paper, we introduce the concept of b -branchings in digraphs, which is a generalization of branchings serving as a counterpart of b -matchings. Here b is a positive integer vector on the vertex set of a digraph D , and a b -branching is defined as a common independent set of two matroids defined by b : an arc set is a b -branching if it has, for every vertex v of D , at most b ( v ) arcs entering v , and it is an independent set of a certain sparsity matroid defined by b and D . We demonstrate that b -branchings yield an appropriate generalization of branchings by extending several classic results on branchings. We first present a multi-phase greedy algorithm for finding a maximum-weight b -branching. We then prove a packing theorem extending Edmonds’ disjoint branchings theorem, and provide a strongly polynomial algorithm for finding optimal disjoint b -branchings. As a consequence of the packing theorem, we prove the integer decomposition property of the b -branching polytope. Finally, we deal with a further generalization in which a matroid constraint is imposed on the b ( v ) arcs sharing the terminal vertex v .

Topics: , Matroid (56%), Disjoint sets (53%), Independent set (52%)

## Summary (1 min read)

### Introduction

• Branchings have several good properties which do not hold for general matroid intersection.
• The objective of this paper is to propose a class of the matroid intersection problem which generalizes branchings and inherits those good properties of branchings.

### 3 Supported by JST CREST Grant Number JPMJCR1402, JSPS KAKENHI Grant Numbers JP16K16012,

• JP26280001, Japan. © N. Kakimura, N. Kamiyama, and K. Takazawa; licensed under Creative Commons License CC-BY 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018).
• This offers a new direction of fundamental extensions of the classical theorems on branchings.
• The authors show that their multi-phase greedy algorithm can be extended to this generalized class.
• In Section 2, the authors review the literature of branchings and matroid intersection, including algorithmic, polyhedral, and packing results.
• The integer decomposition property of the branching polytope is a direct consequence of Theorem 2 and Corollary 4. I Corollary 5 ([1]).

### 3.1 Algorithm description

• The authors present a multi-phase greedy algorithm for finding a maximum-weight b-branching by extending the one for branchings [4, 6, 11, 23].
• The authors first show a key property of Min and Msp, which plays an important role in their algorithm.
• Its proof can be found in the full version [31].
• The authors assume that the arc weights are nonnegative, which are represented by a vector w ∈ RA+.
• It is also straightforward to see that the i-th iteration requires O(|A(i)|) time: Steps 2, 3, and 4 respectively require O(|A(i)|) time.

### 3.2 Optimality of the algorithm and totally dual integral system

• The authors first present a linear program describing MFCS 2018 Algorithm 1 Algorithm bB. Input.
• Finally, as a consequence of their packing theorem, the authors prove the integer decomposition property of the b-branching polytope.
• Its proof is described in the full version [31].
• Optimal matroid intersections, Combinatorial Structures and Their Applications, New York, 1970, Gordon and Breach, 233–234, also known as 39 E.L. Lawler.

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The b-Branching Problem in Digraphs
Naonori Kakimura
1
Keio University, Kanagawa 223-8522, Japan
kakimura@math.keio.ac.jp
Naoyuki Kamiyama
2
Kyushu University and JST, PRESTO, Fukuoka 819-0395, Japan
kamiyama@imi.kyushu-u.ac.jp
Kenjiro Takazawa
3
Hosei University, Tokyo 184-8584, Japan
takazawa@hosei.ac.jp
Abstract
In this paper, we introduce the concept of b-branchings in digraphs, which is a generalization
of branchings serving as a counterpart of b-matchings. Here b is a positive integer vector on
the vertex set of a digraph, and a b-branching is deﬁned as a common independent set of two
matroids deﬁned by b: an arc set is a b-branching if it has at most b(v) arcs sharing the terminal
vertex v, and it is an independent set of a certain sparsity matroid deﬁned by b. We demonstrate
that b-branchings yield an appropriate generalization of branchings by extending several classical
results on branchings. We ﬁrst present a multi-phase greedy algorithm for ﬁnding a maximum-
weight b-branching. We then prove a packing theorem extending Edmonds’ disjoint branchings
theorem, and provide a strongly polynomial algorithm for ﬁnding optimal disjoint b-branchings.
As a consequence of the packing theorem, we prove the integer decomposition property of the b-
branching polytope. Finally, we deal with a further generalization in which a matroid constraint
is imposed on the b(v) arcs sharing the terminal vertex v.
2012 ACM Subject Classiﬁcation Mathematics of computing Graph algorithms
Keywords and phrases Greedy Algorithm, Packing, Matroid Intersection, Sparsity Matroid,
Arborescence
Digital Object Identiﬁer 10.4230/LIPIcs.MFCS.2018.12
Related Version A full version of the paper is available at [31], https://arxiv.org/abs/1802.
02381.
1 Introduction
Since the pioneering work of Edmonds [
12
,
14
], the importance of matroid intersection has
been well appreciated. A special case of matroid intersection is branchings (or arborescences)
in digraphs. Branchings have several good properties which do not hold for general matroid
intersection. The objective of this paper is to propose a class of the matroid intersection
problem which generalizes branchings and inherits those good properties of branchings.
1
Supported by JST ERATO Grant Number JPMJER1201, JSPS KAKENHI Grant Number JP17K00028,
Japan.
2
Supported by JST PRESTO Grant Number JPMJPR14E1, Japan.
3
Supported by JST CREST Grant Number JPMJCR1402, JSPS KAKENHI Grant Numbers JP16K16012,
JP26280001, Japan.
© N. Kakimura, N. Kamiyama, and K. Takazawa;
43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018).
Editors: Igor Potapov, Paul Spirakis, and James Worrell; Article No. 12; pp. 12:1–12:15
Leibniz International Proceedings in Informatics
Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

12:2 The b-Branching Problem in Digraphs
One of the good properties of branchings is that a maximum-weight branching can be
found by a simple combinatorial algorithm [
4
,
6
,
11
,
23
]. This algorithm is much simpler
than general weighted matroid intersection algorithms, and is referred to as a “multi-phase
greedy algorithm” in the textbook by Kleinberg and Tardos [35].
Another good property is the elegant theorem for packing disjoint branchings [
13
]. In
terms of matroid intersection, this theorem says that, if there exist
k
disjoint bases in each
of the two matroids, then there exist
k
disjoint common bases. This packing theorem leads
to a proof that the branching polytope has the integer decomposition property (deﬁned in
Section 2).
In this paper, we propose
b
-branchings, a class of matroid intersection generalizing
branchings, while maintaining the above two good properties. This oﬀers a new direction of
fundamental extensions of the classical theorems on branchings.
Let
D
= (
V, A
) be a digraph and let
b Z
V
++
be a positive integer vector on
V
. For
v V and F A, let δ
F
(v) denote the set of arcs in F entering v, and let d
F
(v) = |δ
F
(v)|.
One matroid M
in
on A has its independent set family I
in
deﬁned by
I
in
= {F A : d
F
(v) b(v) for each v V }. (1)
That is,
M
in
is the direct sum of a uniform matroid on
δ
A
(
v
) of rank
b
(
v
) for every
v V
.
Hence, each vertex can have indegree at most
b
(
v
), which can be more than one. Indeed,
this is the reason why we refer to it as a b-branching, as a counterpart of a b-matching.
In order to make
b
-branchings a satisfying generalization of branchings, the other matroid
should be deﬁned appropriately. Our answer is a sparsity matroid determined by
D
and
b
, which is deﬁned as follows. For
F A
and
X V
, let
F
[
X
] denote the set of arcs in
F
induced by
X
. Also, denote
P
v X
b
(
v
) by
b
(
X
). Now deﬁne a matroid
M
sp
on
A
with
independent set family I
sp
by
I
sp
= {F A : |F [X]| b(X) 1 ( 6= X V )}. (2)
It is known that
M
sp
is a matroid [
20
, Theorem 13.5.1], referred to as a count matroid or a
sparsity matroid.
Now we refer to an arc set
F A
as a b-branching if
F I
in
I
sp
. It is clear that a
branching is a special case of a
b
-branching where
b
(
v
) = 1 for each
v V
. We demonstrate
that
b
-branchings yield a reasonable generalization of branching by proving that the two
fundamental results on branchings can be extended. That is, we present a multi-phase greedy
algorithm for ﬁnding a maximum-weight
b
-branching, and a theorem for packing disjoint
b-branchings.
Our multi-phase greedy algorithm is an extension of the weighted branching algorithm
[
4
,
6
,
11
,
23
], and it has the following features. First, its running time is O(
|V ||A|
), which is
as fast as a simple implementation of the weighted branching algorithm [
4
,
6
,
11
,
23
], and
faster than the current best general weighted matroid intersection algorithm. Second, our
algorithm also ﬁnds an optimal dual solution, which is integer if the arc weights are integer.
Thus, the algorithm constructively proves the total dual integrality of the associated linear
system. Finally, the algorithm leads to a characterization of the existence of a
b
-branching
with prescribed indegree, which is a generalization of that for an arborescence [4, 11, 23].
This characterization theorem is extended to a theorem on packing disjoint
b
-branchings.
Let
k
be a positive integer, and
b
1
, . . . , b
k
be nonnegative integer vectors on
V
such that
b
i
(
v
)
b
(
v
) for each
v V
and
b
i
6
=
b
(
i
= 1
, . . . , k
). Note that, when there exists a
b
-
branching
B
i
satisfying
d
B
i
(
v
) =
b
i
(
v
) for each
v V
(
i
= 1
, . . . , k
), these assumptions about
b
i
follow from the deﬁnition
(1)
and
(2)
of
b
-branchings. We provide a necessary and suﬃcient

N. Kakimura, N. Kamiyama, and K. Takazawa 12:3
condition for
D
to contain
k
disjoint
b
-branchings
B
1
, . . . , B
k
satisfying
d
B
i
(
v
) =
b
i
(
v
) for
every
v V
and
i
= 1
, . . . , k
, which extends Edmonds’ disjoint branching theorem [
13
]. We
then show such disjoint
b
-branchings
B
1
, . . . , B
k
can be found in strongly polynomial time
by at most
|A|
times of submodular function minimization [
28
,
37
,
46
]. We further prove
that, when the arc-weight vector
w R
A
+
is given, disjoint
b
-branchings
B
1
, . . . , B
k
that
minimize
w
(
B
1
) +
· · ·
+
w
(
B
k
) can be found in strongly polynomial time by optimization
over a submodular ﬂow polyhedron [
16
,
21
,
29
,
30
]. By utilizing our disjoint
b
-branchings
theorem, we also prove the integer decomposition property of the b-branching polytope.
We further deal with a generalized class of matroid-restricted
b
-branchings. This is a
special case of matroid intersection in which
M
in
is the direct sum of an arbitrary matroid
on
δ
A
(
v
) of rank
b
(
v
) for all
v V
. Note that, in the class of
b
-branchings, the matroid
M
in
is the direct sum of a uniform matroid on
δ
A
(
v
) of rank
b
(
v
). We show that our multi-phase
greedy algorithm can be extended to this generalized class.
Let us conclude this section with describing related work. The weighted matroid inter-
section problem is a common generalization of various combinatorial optimization problems
such as bipartite matchings, packing spanning trees, and branchings (or arborescences)
in a digraph. The problem has also been applied to various engineering problems, e.g.,
in electric circuit theory [
43
,
44
], rigidity theory [
44
], and network coding [
8
,
25
]. Since
1970s, quite a few algorithms have been proposed for matroid intersection problems, e.g.,
[
5
,
18
,
27
,
37
,
39
,
40
] (See [
26
] for further references). However, all known algorithms are not
greedy, but based on augmentation; repeatedly incrementing a current solution by exchanging
some elements.
The matroids in branchings are a partition matroid and a graphic matroid, which are
interconnected by a given digraph. Such interconnection makes branchings more interesting.
As mentioned before, branchings have properties that matroid intersection of an arbitrary
pair of a partition matroid and a graphic matroid does not have. In particular, extending
the packing theorem of branchings [
13
] is indeed a recent active topic. Kamiyama, Katoh,
and Takizawa [
33
] presented a fundamental extension based on reachability in digraphs,
which is followed by a further extension based on convexity in digraphs due to Fujishige [
22
].
Durand de Gevigney, Nguyen, and Szigeti [
9
] proved a theorem for packing arborescences
with matroid constraints. Király [
34
] generalized the result of [
9
] in the same direction of [
33
].
A matroid-restricted packing of arborescences [
3
,
19
] is another generalization concerning a
matroid constraint. We remark that our packing and matroid restriction for
b
-branchings
diﬀer from the above matroidal extensions of packing of arborescences.
The organization of this paper is as follows. In Section 2, we review the literature of
branchings and matroid intersection, including algorithmic, polyhedral, and packing results.
In Section 3, we present a multi-phase greedy algorithm for ﬁnding a maximum-weight
b
-branching. Section 4 is devoted to proving a theorem on packing disjoint
b
-branchings. In
Section 5, we extend the multi-phase greedy algorithm to matroid-restricted
b
-branchings.
In Section 6, we conclude this paper with a couple of remarks.
2 Preliminaries
In this section, we review fundamental results on branchings and related theory of matroid
intersection and polyhedral combinatorics. For more details, refer to [32, 36, 47].
In a digraph
D
= (
V, A
), an arc subset
B A
is a branching if, in the subgraph (
V, B
),
the indegree of every vertex is at most one and there does not exist a cycle in the undirected
sense. In terms of matroid intersection, a branching is a common independent set of a
M F C S 2 0 1 8

12:4 The b-Branching Problem in Digraphs
partition matroid and a graphic matroid, i.e., intersection of
{F A : d
F
(v) 1 for each v V }, (3)
{F A : |F [X]| |X| 1 ( 6= X V )}. (4)
Recall that a branching is a special case of a
b
-branching where
b
(
v
) = 1 for each
v V
.
Indeed, by putting
b
(
v
) = 1 for each
v V
in
(1)
and
(2)
, we obtain
(3)
and
(4)
, respectively.
As stated in Section 1, a maximum-weight branching can be found by a multi-phase
greedy algorithm [
4
,
6
,
11
,
23
], which appears in standard textbooks such as [
35
,
36
,
47
]. To
the best of our knowledge, we have no other nontrivial special case of matroid intersection
which can be solved greedily. For example, intersection of two partition matroids is equivalent
to bipartite matching. This seems the simplest nontrivial example of matroid intersection,
but we do not know a greedy algorithm for ﬁnding a maximum bipartite matching.
Another important result on branchings is the disjoint branchings theorem by Edmonds
[
13
], described as follows. For a positive integer
k
, the set of integers
{
1
, . . . , k}
is denoted
by [
k
]. For
F A
and
X V
, let
δ
F
(
X
)
A
denote the set of arcs in
F
from
V \ X
to
X
,
and let d
F
(X) = |δ
F
(X)|.
I Theorem 1
(Edmonds [
13
])
.
Let
D
= (
V, A
) be a digraph and
k
be a positive integer,
and
U
1
, . . . , U
k
be subsets of
V
. Then, there exist disjoint branchings
B
1
, . . . , B
k
such that
U
i
= {v V : d
B
i
(v) = 1} for each i [k] if and only if
d
A
(X) |{i [k]: X U
i
}| ( 6= X V ).
From Theorem 1, we obtain a theorem on covering a digraph by branchings [17, 41].
I Theorem 2
([
17
,
41
])
.
Let
D
= (
V, A
) be a digraph and let
k
be a nonnegative integer.
Then, the arc set A can be covered by k branchings if and only if
d
A
(v) k (v V ),
|A[X]| k(|X| 1) ( 6= X V ).
Theorem 2 leads to the integer decomposition property of the branching polytope. The
branching polytope is a convex hull of the characteristic vectors of all branchings. It follows
from the total dual integrality of matroid intersection [
12
] that the branching polytope is
determined by the following linear system:
x(δ
(v)) 1 (v V ), (5)
x(A[X]) |X| 1 ( 6= X V ), (6)
x(a) 0 (a A). (7)
I Theorem 3 (see [47]). The linear system (5)(7) is totally dual integral.
I Corollary 4 (see [47]). The linear system (5)(7) determines the branching polytope.
For a polytope
P
and a positive integer
k
, deﬁne
kP
=
{x: x
0
P, x
=
kx
0
}
. A polytope
P
has the integer decomposition property if, for each positive integer
k
, any integer vector
x kP
can be represented as the sum of
k
integer vectors in
P
. The integer decomposition
property of the branching polytope is a direct consequence of Theorem 2 and Corollary 4.
I Corollary 5 ([1]). The branching polytope has the integer decomposition property.

N. Kakimura, N. Kamiyama, and K. Takazawa 12:5
We remark that the integer decomposition property does not hold for an arbitrary matroid
intersection polytope. Schrijver [
47
] presents an example of matroid intersection deﬁned
on the edge set of
K
4
without integer decomposition property. Indeed, ﬁnding a class of
polyhedra with integer decomposition property is a classical topic in combinatorics. Typical
examples of polyhedra with integer decomposition property include polymatroids [
1
,
24
],
the branching polytope [
1
], and intersection of two strongly base orderable matroids [
7
,
42
].
While there is some recent progress [
2
], the integer decomposition property of polyhedra is
far from being well understood. In Section 4, we will prove that the
b
-branching polytope is
a new example of polytopes with integer decomposition property.
3 Multi-phase greedy algorithm
3.1 Algorithm description
In this subsection, we present a multi-phase greedy algorithm for ﬁnding a maximum-weight
b
-branching by extending the one for branchings [
4
,
6
,
11
,
23
]. Let
D
= (
V, A
) be a digraph
and
b Z
V
++
be a positive integer vector on
V
. Recall that an arc set
F A
is a
b
-branching
if F I
in
I
sp
, where I
in
and I
sp
are deﬁned by (1) and (2), respectively.
We ﬁrst show a key property of
M
in
and
M
sp
, which plays an important role in our
algorithm. Its proof can be found in the full version [31].
I Lemma 6.
An independent set
F
in
M
in
is not independent in
M
sp
if and only if (
V, F
)
has a strong component X such that
|F [X]| = b(X). (8)
Moreover, for every strong component X in (V, F ) satisfying (8), F [X] is a circuit of M
sp
.
Lemma 6 enables us to design the following multi-phase greedy algorithm for ﬁnding a
maximum-weight b-branching:
Find a maximum-weight independent set F in M
in
.
If (
V, F
) has a strong component
X
satisfying
(8)
, then contract
X
, reset
b
and the
weights of the remaining arcs appropriately, and recurse.
At the end of the algorithm, we expand every contracted component
X
in the following
manner. Suppose that the solution
F
has an arc
a
0
entering the vertex
v
X
created when
contracting
X
. Denote the terminal vertex of
a
0
before contracting
X
by
v
0
X
. In
expanding
X
b
(
X
)
1 arcs to
F
, consisting of
b
(
v
) heaviest arcs among
δ
A
(
v
)
A
[
X
]
for each
v X \{v
0
}
and
b
(
v
0
)
1 heaviest arcs among
δ
A
(
v
0
)
A
[
X
]. If
F
has no arc entering
v
X
, then we add
b
(
X
)
1 arcs to
F
, consisting of
b
(
v
) heaviest arcs among
δ
A
(
v
)
A
[
X
] for
each v X except for the arc of minimum weight among those b(X) arcs.
A formal description of the algorithm is as follows. We denote an arc a A with initial
vertex
u
and terminal vertex
v
by (
u, v
). We assume that the arc weights are nonnegative,
which are represented by a vector w R
A
+
. For F A, we denote w(F ) =
P
aF
w(a).
The complexity of Algorithm
b
B is analyzed as follows. It is clear that there are at most
|V |
iterations. It is also straightforward to see that the
i
-th iteration requires O(
|A
(i)
|
) time:
Steps 2, 3, and 4 respectively require O(
|A
(i)
|
) time. Thus, the total time complexity of the
algorithms is O(|V ||A|).
3.2 Optimality of the algorithm and totally dual integral system
In this subsection, we prove that the output of Algorithm
b
B is a maximum-weight
b
-
branching by the following primal-dual argument. We ﬁrst present a linear program describing
M F C S 2 0 1 8

##### Citations
More filters

Book ChapterDOI
04 May 2020
TL;DR: This paper presents two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into b-branchings in Digraphs, and proves that equitable partitions always exist.
Abstract: An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into k branchings, there always exists an equitable partition into k branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into b-branchings in digraphs. For matching forests, Kiraly and Yokoi (2018) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of the covered vertices. For b-branchings, we define an equitability based on the size of the b-branching and the indegrees of all vertices. For both matching forests and b-branchings, we prove that equitable partitions always exist.

3 citations

Posted Content
TL;DR: A linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation for \$b-bibranchings are presented, which imply polynomial algorithms for finding a shortest bibranching.
Abstract: In this paper, we introduce the \$b\$-bibranching problem in digraphs, which is a common generalization of the bibranching and \$b\$-branching problems The bibranching problem, introduced by Schrijver (1982), is a common generalization of the branching and bipartite edge cover problems Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation The \$b\$-branching problem, recently introduced by Kakimura, Kamiyama, and Takazawa (2018), is a generalization of the branching problem admitting higher indegree, ie, each vertex \$v\$ can have indegree at most \$b(v)\$ For \$b\$-branchings, a combinatorial algorithm, a linear programming formulation with total dual integrality, and a packing theorem for branchings are extended A main contribution of this paper is to extend those previous results on bibranchings and \$b\$-branchings to \$b\$-bibranchings That is, we present a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation for \$b\$-bibranchings In particular, the linear program and M-convex submodular flow formulations respectively imply polynomial algorithms for finding a shortest \$b\$-bibranching

3 citations

### Cites background or methods from "The b-branching problem in digraphs..."

• ...A theorem on packing disjoint b-branchings is also presented in [13], which extends Edmonds’ disjoint branchings theorem [6] and leads to the integer decomposition property of the b-branching polytope....

[...]

• ...Second, the proof for the packing theorem utilizes the disjoint b-branchings theorem [13] and the supermodular coloring theorem [22], the latter of which was used in an alternative proof [22] for the packing theorem for bibranchings....

[...]

• ...[13] presented a multi-phase greedy algorithm for finding a longest b-branching, which extends that for branchings [2–4, 9]....

[...]

• ...In our formulation, this computation amounts to computing the minimum weight of a b-branching with prescribed indegree, which can be done by using a combinatorial algorithm for the longest b-branching [13]....

[...]

• ...Finally, for the M-convex submodular flow formulation, we newly prove an exchange property of b-branchings, which extends that for branchings [23] and follows from the disjoint b-branchings theorem [13]....

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Posted Content
TL;DR: This paper presents two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into \$b-branchings in Digraphs, and proves that equitable partitions always exist.
Abstract: An equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into \$k\$ branchings, there always exists an equitable partition into \$k\$ branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into \$b\$-branchings in digraphs. For matching forests, Kiraly and Yokoi (2018) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of the covered vertices. For \$b\$-branchings, we define an equitability based on the size of the \$b\$-branching and the indegrees of all vertices. For both matching forests and \$b\$-branchings, we prove that equitable partitions always exist.

2 citations

Journal ArticleDOI
23 Mar 2021-Networks
Abstract: In this paper, we introduce the \$b\$-bibranching problem in digraphs, which is a common generalization of the bibranching and \$b\$-branching problems. The bibranching problem, introduced by Schrijver (1982), is a common generalization of the branching and bipartite edge cover problems. Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation. The \$b\$-branching problem, recently introduced by Kakimura, Kamiyama, and Takazawa (2018), is a generalization of the branching problem admitting higher indegree, i.e., each vertex \$v\$ can have indegree at most \$b(v)\$. For \$b\$-branchings, a combinatorial algorithm, a linear programming formulation with total dual integrality, and a packing theorem for branchings are extended. A main contribution of this paper is to extend those previous results on bibranchings and \$b\$-branchings to \$b\$-bibranchings. That is, we present a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation for \$b\$-bibranchings. In particular, the linear program and M-convex submodular flow formulations respectively imply polynomial algorithms for finding a shortest \$b\$-bibranching.

2 citations

Proceedings ArticleDOI
01 Aug 2018
TL;DR: This paper presents a multi-phase greedy algorithm for finding a maximum-weight \$b-branching, proves a packing theorem extending Edmonds' disjoint branchings theorem, and provides a strongly polynomial algorithm forFinding optimal disjointed \$b\$-br branchings.
Abstract: In this paper, we introduce the concept of b -branchings in digraphs, which is a generalization of branchings serving as a counterpart of b -matchings. Here b is a positive integer vector on the vertex set of a digraph D , and a b -branching is defined as a common independent set of two matroids defined by b : an arc set is a b -branching if it has, for every vertex v of D , at most b ( v ) arcs entering v , and it is an independent set of a certain sparsity matroid defined by b and D . We demonstrate that b -branchings yield an appropriate generalization of branchings by extending several classic results on branchings. We first present a multi-phase greedy algorithm for finding a maximum-weight b -branching. We then prove a packing theorem extending Edmonds’ disjoint branchings theorem, and provide a strongly polynomial algorithm for finding optimal disjoint b -branchings. As a consequence of the packing theorem, we prove the integer decomposition property of the b -branching polytope. Finally, we deal with a further generalization in which a matroid constraint is imposed on the b ( v ) arcs sharing the terminal vertex v .

##### References
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TL;DR: This book shows the combinatorial optimization polyhedra and efficiency as your friend in spending the time in reading a book.
Abstract: Reading a book is also kind of better solution when you have no enough money or time to get your own adventure. This is one of the reasons we show the combinatorial optimization polyhedra and efficiency as your friend in spending the time. For more representative collections, this book not only offers it's strategically book resource. It can be a good friend, really good friend with much knowledge.

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TL;DR: Using F-heaps, a new data structure for implementing heaps that extends the binomial queues proposed by Vuillemin and studied further by Brown, the improved bound for minimum spanning trees is the most striking.
Abstract: In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in O(log n) amortized time and all other standard heap operations in O(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms. In particular, we obtain the following worst-case bounds, where n is the number of vertices and m the number of edges in the problem graph: O(n log n + m) for the single-source shortest path problem with nonnegative edge lengths, improved from O(mlog(m/n+2)n);O(n2log n + nm) for the all-pairs shortest path problem, improved from O(nm log(m/n+2)n);O(n2log n + nm) for the assignment problem (weighted bipartite matching), improved from O(nmlog(m/n+2)n);O(mβ(m, n)) for the minimum spanning tree problem, improved from O(mlog log(m/n+2)n); where β(m, n) = min {i | log(i)n ≤ m/n}. Note that β(m, n) ≤ log*n if m ≥ n.Of these results, the improved bound for minimum spanning trees is the most striking, although all the results give asymptotic improvements for graphs of appropriate densities.

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Abstract: This comprehensive textbook on combinatorial optimization places special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. It has arisen as the basis of several courses on combinatorial optimization and more special topics at graduate level. It contains complete but concise proofs, also for many deep results, some of which did not appear in a textbook before. Many very recent topics are covered as well, and many references are provided. Thus this book represents the state of the art of combinatorial optimization. This fourth edition is again significantly extended, most notably with new material on linear programming, the network simplex algorithm, and the max-cut problem. Many further additions and updates are included as well. From the reviews of the previous editions: "This book on combinatorial optimization is a beautiful example of the ideal textbook." Operations Research Letters 33 (2005), p.216-217 "The second edition (with corrections and many updates) of this very recommendable book documents the relevant knowledge on combinatorial optimization and records those problems and algorithms that define this discipline today. To read this is very stimulating for all the researchers, practitioners, and students interested in combinatorial optimization." OR News 19 (2003), p.42 "... has become a standard textbook in the field." Zentralblatt MATH 1099.90054

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TL;DR: Algorithm Design introduces algorithms by looking at the real-world problems that motivate them and encourages an understanding of the algorithm design process and an appreciation of the role of algorithms in the broader field of computer science.
Abstract: Algorithm Design introduces algorithms by looking at the real-world problems that motivate them. The book teaches students a range of design and analysis techniques for problems that arise in computing applications. The text encourages an understanding of the algorithm design process and an appreciation of the role of algorithms in the broader field of computer science.

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Abstract: The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all bases have the same cardinality. (See Van der Waerden, Section 33.)

888 citations

##### Frequently Asked Questions (1)
###### Q1. What are the contributions in "The b-branching problem in digraphs" ?

In this paper, the authors introduce the concept of b-branchings in digraphs, which is a generalization of branchings serving as a counterpart of b-matchings. The authors demonstrate that b-branchings yield an appropriate generalization of branchings by extending several classical results on branchings. The authors first present a multi-phase greedy algorithm for finding a maximumweight b-branching. The authors then prove a packing theorem extending Edmonds ’ disjoint branchings theorem, and provide a strongly polynomial algorithm for finding optimal disjoint b-branchings. As a consequence of the packing theorem, the authors prove the integer decomposition property of the bbranching polytope. Finally, the authors deal with a further generalization in which a matroid constraint is imposed on the b ( v ) arcs sharing the terminal vertex v. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms