BOUNDED-DEGREE ROOTED TREE AND TDI-NESS

. This paper contributes to the polyhedral aspect of the Maximum-Weight Bounded-Degree Rooted Tree Problem, where only edge-indexed variables are considered. An initial formulation is given, followed by an analysis of the dimension and a facial study for the polytope. Several families of new valid inequalities are proposed, which enables us to characterize the polytope on trees and cycles with a totally dual integral system.


Introduction
Given an undirected graph  with node set  and edge set , and a specified node  of  , hereafter called the root, a rooted tree is either the empty graph (∅, ∅) or a tree (i.e., a connected and acyclic subgraph) of  containing node .If a positive integer   is associated with each node  of  , then a rooted tree  of  is called bounded-degree whenever the degree of each node  in  does not exceed its degree requirement, or capacity,   .This paper deals with the polyhedral structure of the Bounded-Degree Rooted Tree (BDRT) polytope, that is, the convex hull of the incidence vectors of edge sets inducing bounded-degree rooted trees of .
To the best of our knowledge, this polytope has not been previously studied in the literature.Actually, the problem of considering bounded-degree rooted trees quite recently arises in the delivery of video streams in under-provisioned peer-to-peer networks, where the scarce resources lie at the peers' level (e.g., average available upload capacity below the stream bit-rate) and not at the links' one [1].Such peer-to-peer networks usually are represented by non-necessarily complete graphs (due to peering agreements, too-long transmission delays, or too-high jitters) where the video stream's source naturally corresponds to the root and for each peer, its upload-capacity limit can be converted into an upper bound on the number of peers it can send the video stream to [1].The problem, considered in [1] and called the Maximum Bounded-Degree Rooted Tree (MBDRT) problem, then consists of finding a rooted tree which respects the degree constraints and maximizes the number of nodes it contains.In [14], the MBDRT problem was showed to be an NP-hard combinatorial optimization problem by reducing the 3-SAT problem [9] to it, and polynomial-time algorithms were given on certain classes of graphs such as trees, cycles and complete graphs.
If a real-value edge-weight vector is given, the Maximum-Weight Bounded-Degree Rooted Tree Problem, hereafter denoted MWBDRTP, consists of finding a maximum-weight subset of edges which induces a boundeddegree rooted tree of .The NP-hardness of MWBDRTP is twofold since it comes from both the degree restriction and the non-spanning property.
On the one hand, bounded-degree versions of combinatorial optimization problems have received a significant amount of research interest over the last two decades.Several approximation algorithms have been devised for the bounded-degree spanning tree problem (see [18] and the references therein), the bounded-degree Steiner tree problem [8], and bounded-degree matroids and submodular flows [15], but no intensive polyhedral studies seem to exist.Most of these algorithms are based on polyhedral combinatorics [17] in the sense that they use a linear relaxation for the problem to first provide a dual bound and then use the optimal solution to this relaxation to generate a primal solution.
On the other hand, combinatorial optimization problems with the non-spanning property, such as the Steiner tree problem (see [4] and the references therein), the survivable network design problem [12], and the maximumweight edge-induced connected subgraph problem [5], have been considered in the literature for several decades.The polyhedral structures of these problems have been intensively studied (see [4,5,12] and the references therein) and some linear-relaxation based approximation algorithms have been designed (see, e.g., [11]).Besides, the feasibility problem of MWBDRTP is not NP-complete, contrary to the case for other bounded-degree problems such as bounded-degree spanning tree problem and bounded-degree Steiner tree problem.
This paper explores the polyhedral structure of MWBDRTP, and is organized as follows.In Section 2, the Bounded-Degree Rooted Tree (BDRT) polytope, denoted ℬ(, , c), is defined and its dimension is studied.An initial formulation is provided, followed by some necessary and sufficient conditions for the inequalities in the formulation to be facet-defining.Then in Section 3, two families of new valid inequalities are introduced.By using these new valid inequalities, a complete polyhedral characterization of ℬ(, , c) is given in Section 4 on either trees and cycles.In fact, the proposed linear systems not only lead to integral polytopes but also are Totally Dual Integral (TDI) [17].
This introduction is concluded with some definitions and notation, which have been mainly taken from [6,17].
Let  be a simple, connected, and undirected graph with node set  () and edge set (); when there is no confusion on the graph from the context, the graph is labeled as  = (, ).If  ∈  is an edge with extremities  and ,  is also used to denote .
Let  be a subset of  .The set of edges having one extremity in  and the other one in  =  ∖  is called a cut and is denoted by ( ).If  = {} for some  ∈  , then we write () for ({}).We denote by [ ] the set of edges having both extremities in  , and [ ] the subgraph induced by  (i.e., [ ] = (, [ ])).Similarly, given  ⊆ ,  [ ] is used to denote the node set composed of extremities of edges in  , and [ ] the subgraph induced by  (i.e., [ ] = ( [ ],  )).Given two sets of nodes  and  with  ⊆  ∖  , the set of edges having one extremity in  and the other one in  is denoted by [ is a partition of  , then we denote by () the set of edges having their extremities in different classes of .For any node  ∈  , let  () ⊆  denote the set of neighbours of  in .Besides, in this paper we represent a path by its edge set.For any node  ∈  , let   denote a path between  and .  can also be refereed to as an - path.Similarly, for any edge  ∈ , let   denote a path between  and , and   is also refereed to as an - path.Given any edge subset  ⊆ , its incidence vector is the vector x  in {0, 1}  such that    = 1 if and only if  ∈  .Given any vector x ∈ R  and any edge set  ⊆ , x( ) is used for ∑︀ ∈   .Of all the nodes of , we need to distinguish the non-root nodes having unit degree requirements from others, for any of the former, if present in a bounded-degree rooted tree of , always appears as a leaf.Thus the set of these unit-capacity nodes is denoted by A node  (edge , respectively) of  is unreachable from root  if there does not exist any bounded-degree - path (- path, respectively) in  or equivalently, each - path (- path, respectively) in  contains an inner node in .Let   and   be the sets composed of the unreachable nodes and edges of , respectively.Notice that there might exist edges in   whose extremities do not belong to   , that is, {() :  ∈   } ⊆   .
Solving MBDRT problem on  can hence be reduced to solving MBDRT problem on  ′ = ( ∖   ,  ∖   ), the graph obtained from  by deleting both unreachable nodes and edges.Notice that getting rid of the unreachable elements in a graph can be performed in linear time by various search algorithms.We therefore make the following assumption for the remainder of the paper.Assumption 1.1.Graph  contains no unreachable nodes or edges.

The BDRT polytope
Let  ⊆ {0, 1}  be the set composed of all the incidence vectors of the edge sets inducing bounded-degree rooted trees of , that is, The BDRT polytope hence is the convex hull of  and hereafter is denoted ℬ(, , c).The initially proposed formulation consists of the following inequalities.
The connectivity inequalities (2.1) guarantee that each selected edge is connected to root  through a path.
The well-known subtour elimination inequalities (2.2) ensure that there is no cycles in the resulting graph [3].
The degree requirement imposed on each node is handled by the capacity inequalities (2.3).The bounds on the variables are guaranteed by the trivial box inequalities (2.4) and (2.5).These aforementioned inequalities clearly give a formulation for ℬ(, , c), or equivalently they induce a polytope  (, , c) whose integer hull is ℬ(, , c).

Dimension
We first establish a technical lemma that will come in handy in the forthcoming proofs of the dimension and the facets of ℬ(, , c).Lemma 2.2.Given an undirected and connected graph  = (, ), a node  ∈  , and a node-capacity vector c, let  be any nonempty subset of edges of .For any  ∈  , consider an - path   in  that satisfies the capacity constraints and contains as few edges as possible.The vectors in the set {x  :  ∈  } are affinely independent.
It is worth noting that here we intentionally avoid the notion of shortest path between  and  (or ) in , since a shortest path in  does not necessarily satisfy the capacity constraints due to the nodes of .Combining all these non-zero affinely independent vectors of  with the zero vector gives a set of || + 1 affinely independent vectors, each of which inducing a bounded-degree rooted tree of .One therefore has dim ℬ(, , c) ≥ || and our proof is complete for || is a trivial upper bound on dim ℬ(, , c).
If Assumption 1.1 was dropped, we would clearly have dim ℬ(, , c) = || − |  | for any incidence vector of the edge set of a bounded-degree rooted tree of  straightforwardly would satisfy the following equations   = 0 for all  ∈   . ( Moreover there would exist a one-to-one correspondence between the facet-defining inequalities of ℬ(, , c) and those of ℬ( ′ , , c ′ ), where c ′ is the restriction of c to  ′ .In fact, any facet-defining inequality a  x ≤  of ℬ(, , c) could be written as a ′  x ′ + ∑︀ ∈     ≤ , where x ′ is the restriction of x to  ′ , a ′  x ′ ≤  is a facet-defining inequality of ℬ( ′ , , c ′ ), and  ∈ R  .Consequently in terms of polyhedral characterizations of BDRT polytope, any complete polyhedral characterization of ℬ(, , c) could easily be deduced from any of ℬ( ′ , , c ′ ), and vice-versa.

Properties of Facet-defining inequalities
Necessary and sufficient conditions for inequalities (2.1)-(2.5) to be facet-defining of ℬ(, , c) have been established.We refer to [20] for detailed statements of these conditions and for their proofs.The latter follow standard techniques and are based on the following general properties facet-defining inequalities of ℬ(, , c) must satisfy.These properties will hereafter be used to devise new facet-defining inequalities and manage redundancy in ℬ(, , c) on trees and cycles.
The following lemma describes a property related to the mandatory nonnegative coefficients of the edges of () in any facet-defining inequalities.
Lemma 2.4.Let a  x ≤  be a valid inequality for ℬ(, , c) different from a negative scalar multiple of any nonnegativity inequality (2.5).Inequality a  x ≤  is facet-defining of ℬ(, , c) only if   ≥ 0 for any edge  ∈ () or any pendant edge  of .
Proof.Suppose that a  x ≤  defines a facet ℱ of ℬ(, , c) and there exists an edge   ∈ () such that   < 0. (The proof is similar if   is a pendant edge of .) Theorem 2.3, combined with the assumption on a  x ≤ , implies ℱ ̸ = {x ∈ ℬ(, , c) :   = 0} for all  ∈ .There then must exist an edge set  ⊆  such that   ∈  , [ ] is a bounded-degree rooted tree of , and A necessary and sufficient condition for the root node's capacity inequality to be facet-defining was given in [20] as stated in the next proposition.
The capacity   of the root node impacts the possible values the right-hand sides of facet-defining inequalities take.If the root node has unit capacity, then the right-hand side of any facet-defining inequality of ℬ(, , c) but x(()) ≤   must equal 0 as stated in the following lemma.
Lemma 2.6.Let a  x ≤  be a valid inequality for ℬ(, , c) different from a negative scalar multiple of inequality (2.7).
As   = 1 any bounded-degree rooted tree of  whose edge set is nonempty satisfies Besides, for those inequalities having only non-negative coefficients, the following lemma can be developed.Lemma 2.7.Given a valid inequality ax ≤  for ℬ(, , c) with a ≥ 0, a ̸ = 0,  > 0, let (1) there does not exist an edge   ∈  such that it is in each bounded-degree - path for any edge  ∈  + , unless  + = {  }; (2) there does not exist a node   ∈  ∖ {} with   = 2 such that it is an inner node in each bounded-degree − path for any edge  ∈  + .
Proof.Suppose that there exists a valid inequality ax ≤  and an edge   ∈  such that the first condition is not satisfied.Consider any  ⊆  with its incidence vector x  that satisfies ax =  > 0. One must have  + ∈  for some  + ∈  + .Since [ ] is a bounded-degree rooted tree, it must contain a bounded-degree − + , which has to include   .Hence   ∈  , which implies the inequality    ≤ 1 induces a larger face than the one defined by ax ≤ .Now suppose there exists a node   ∈  ∖ {} that is an inner node in every bounded-degree − path for any edge  ∈  + .For any  ⊆  with its incidence vector x  satisfying ax = , x((  )) =   is also satisfied.As a result, x((  )) ≤   induces a larger face than the one defined by ax ≤ .
This lemma expresses that in the circumstances where the coefficients are non-negative and the right-hand side of a facet-defining inequality are positive, the associated graph does not contain certain substructures, specifically the bridges or articulation nodes with properties described above.It is worth noting that here the notion of bridges and articulation nodes needs to respect the capacity factor.For instance, an edge might not be a bridge in the graph, but regarding capacity, removing it might lead to the removal of all bounded-degree paths between  and some other edges.In this case, it can be deemed as a bridge regarding capacity.
To illustrate how these lemmas are reflected on previously introduced valid inequalities, we hereafter give an example.According to Lemma 2.4, any connectivity inequality associated with a set  ⊆  ∖ {} that satisfies () ∩ () ̸ = ∅ is not facet-defining, as stated in the following proposition.(2.8) It is worth mentioning that (2.8) covers certain facets which (2.1) does not.Hence, in latter discussion (2.8) is always considered instead of (2.1).Necessary and sufficient conditions for each family of the inequalities (2.2)-(2.8)and their detailed proof can be found in [20].

New valid inequalities
Besides the inequalities introduced previously, there are a few sets of new constraints that have been discovered during our work. Let The pair (, ) is called a rooted matching-partition of .The concept of matching-partition originates from [5] for the connected subgraph problem.
Let ℳ() denote the set composed of all the rooted matching-partitions of , and denote by () the set of edges having their extremities in different classes of partition .With any rooted matching-partition (, ) ∈ ℳ(), one can associate the following rooted matching-partition inequality to connect all the edge in  ∩  with .Hence, (,  ) is not connected, which forms a contradiction.
The rooted matching-partition inequalities introduced here are different from those proposed by [5], mainly due to the existence of the root and the capacity constraints.
It is worth noting that the rooted matching-partition inequalities generalizes the connectivity inequalities.Particularly, the connectivity inequalities can be seen as a special case of the rooted matching-partition inequalities where one always has | | = 1.Thus, in the facial study we focus on the cases with | | ≥ 2.
In order to facilitate the forthcoming discussion on the facial study results of rooted matching-partition inequalities, some definition needs to be introduced beforehand.Given (, ) ∈ ℳ(), let  ′  be the graph obtained from  by first removing () ∩ () and then shrinking each   ∈  into a node, and each non-empty edge set [  ,   ] ∖ () ⊆ () ∖ () into an edge, for any distinct ,  ∈ {1, • • • , }.The following theorem gives the necessary and sufficient facet-defining conditions for the rooted matching-partition inequalities.The proof of the necessity and sufficiency of the conditions can be found in [20].For each of these conditions, if it were not satisfies, one could always construct another valid inequality that would induce a larger face of ℬ(, , c).For instance, if condition (1) or (2) were not satisfied, one could find a lower bound inequality (2.5) or another distinct matching-partition inequality that would induce a larger face of ℬ(, , c).Conversely, for the sufficiency, we prove that if all the conditions are satisfied, the face induced by the matching-partition inequality is not a proper face of any other proper faces of ℬ(, , c).
Besides the rooted matching-partition inequalities, another new family of inequalities, the upload-capacity inequalities, are also found to be facet-defining for ℬ(, , c).Given a node set  ⊂  ∖ {} and a node  in  ∖ , the upload-capacity inequality is defined as follows.
x(()) The necessary and sufficient conditions for the upload-capacity inequalities to be facet-defining is described in the following proposition.Similar to the case of matching-partition inequalities, for each of these conditions, if it is violated, one can always construct another valid inequality that induces a larger face of ℬ(, , c).For instance, if condition (1) or ( 2) is violated, one can find a connectivity inequality or a lower bound inequality that induces a larger face of ℬ(, , c).Conversely, for the sufficiency, we prove that if all the conditions are satisfied, the face induced by the upload-capacity inequality is not a proper face of any other faces.
For the nodes in , the presentation of upload-capacity inequalities is slightly different.Given a node set  ⊆  ∖ {} with   ∈  ∩ , the upload-capacity inequality associated with  and   is as follows.

x(𝛿(𝑣
Its validity can also be proved and is stated in the following theorem. Theorem 3.5.Inequality (3.3) is valid for ℬ(, , c).
(2) there does not exist an edge  ∈ [] such that removing  ∪ ( ∖   ) from  disconnects  and   ; (3) there does not exist a node  ∈  ∖ {  } with   = 2 such that removing () ∪ ( ∖   ) from  disconnects  and   ; The proof for the necessary and sufficient conditions is similar to the upload-capacity inequalities associated with  ∈  ∖ .Detailed proof can be found in [20].
These two aforementioned families of inequalities play an important role in the characterization of ℬ(, , ) on trees and cycles.

Characterizations of ℬ(𝐺, 𝑐, 𝑟) and TDI-ness
In this section, we show that with the rooted matching-partition inequalities and upload-capacity inequalities being introduced, ℬ(, , c) can be characterized on trees and cycles with TDI systems.

Primal formulation and subproblems
According to the aforementioned results on valid inequalities and their facet-defining conditions, one can deduce that some of these inequalities is redundant.After getting rid of the redundant inequalities, one can get the following formulation for ℬ(, , c) on trees, max wx x(()) −     ≤ 0 for all  ∈  ∖ {}, (4.2) x(()) ≤   , (4.3) ≥ 0 for all  is a leaf edge, ( where   denotes the edge in () and also in the - path   for  ∈  ∖ {} (i.e.,   ∈ () ∩   ), and   denotes the edge that is adjacent to  and also in the - path   for  ∈  ∖ ().Note that inequalities (4.1) and inequalities(4.2) are special cases of connectivity inequalities (2.8) and upload capacity inequalities (3.2) respectively.Let the polytope defined by the linear system composed of (4.1)-(4.5)be   (, , c) = {x ∈ R  : x satisfies (4.1)-(4.5)}.
We hereafter show that it is a ideal formulation for ℬ(, , c) on trees and that the linear system defining   (, , c) is TDI.Note that since TDI-ness is a sufficient condition for integrality [7], the integrality of   (, , c) could be seen as a direct consequence of the next theorem.
Consider the linear program max{wx : where w ∈ R  .Theorem 4.1 is proved by showing that one can always obtain an optimal dual solution to (4.6) and this solution is integral if w ∈ Z  .We break the proof of Theorem 4.1 into several technical lemmas and propositions, and then provide a proof at the end of this section.
Given any node  ∈  , let () be the value of a maximum bounded-degree tree rooted at  of the subgraph [⌊⌋] (of  induced by ⌊⌋, the up-closure of ), where the capacity vector c  ∈ Z ⌊⌋ + satisfies In other words, () = max{x( ) : [ ] is a tree of [⌊⌋] rooted at  and bounded by c  }.
The following technical lemma is given as a support of our later results.
Proof.Suppose otherwise that a maximum bounded-degree tree   rooted at  contains a bounded-degree tree    rooted at   which is not maximum.By replacing    in   by a maximum bounded-degree tree rooted at   , one obviously obtains a bounded-degree tree rooted at  whose weight is larger than   .Hence, it contradicts with the assumption.
Effectively, Lemma 4.2 reduces the MWBDRTP to a series of subproblems, which can be solved with a dynamic programming approach.Details on a dynamic programming algorithm proposed for MWBDRTP on trees can be found in [20].In this paper, we emphasize on the algorithm that obtains the dual solution.
The following part provides some notation and parameters that will be crucial in the TDI-ness proof.Given a non-leaf node  ∈  , for any edge   ∈  with   ∈  () ∩ ⌊⌋, we define a function According to Lemma 4.2, the problem of calculating () reduces to max As it is a maximization problem over a uniform matroid if  (︀   )︀ is known for all  ∈ {1, • • • ,   }, it can be easily solved by a greedy algorithm in linear time, where at each step one selects a node   with the maximum non-negative ℎ (︀   )︀ until there is no such nodes or    nodes have been selected.Without loss of generality, assume that ℎ( The following equation holds.Hereafter we provide a solution to the dual program, and prove its feasibility and optimality.First of all, the value of  can be first decided as follows By the definition of   , one can deduce that   ≥ 0 for any  ∈  .For any edge  =     , let Note that for any leaf edge     ∈ , since   = 0 one has Consider a non-leaf node  ∈  .By the definition of   ,   ≤    holds, whereas from the definition of   , one also has   = 0 if   <    .Thus always holds.For any  >   , we have that ℎ(  ) ≤ ℎ(  ) =   if   =    , whereas ℎ(  ) ≤ 0 =   if   <    .Hence ℎ(  ) ≤   holds and thus Similarly, for any  ≤   it can be deduced from ℎ(  ) ≥   that Therefore, for any non-leaf node  ∈  , we have where the first equality comes from (4.8), the second equality is from (4.15) and (4.16), and the last equality is from (4.14).
Immediately, for any non-leaf edge  =     ∈ , one has Another result one can deduce from (4.17) is Now we can construct a solution (, ) based on ( ′ , ), and then prove that it is dual-feasible, optimal, and is integral if w is integral.
For each non-leaf edge  =     ∈ , the difference between the left-hand side and right-hand side of the dual constraint (4.9) associated with  and ( ′ , ) is denoted as Let the set of non-leaf edges that satisfy Now we show that there exists a vector ∆ ∈ R  + such that  =  ′ + ∆ and (, ) is dual-feasible and optimal.Algorithm 1 computes the vector ∆.
The feasibility and optimality of (, ) can then be proved.Proof.For each non-leaf edge  =     , one clearly has For any non-leaf edge  =     , Algorithm 1 guarantees that Algorithm 1: Algorithm on trees to obtain ∆.

4
For each edge in  ′ ∈
In addition, for any  ∈ ,  ′  , ∆  ≥ 0 leads to   ≥ 0. Therefore, (, ) is dual-feasible.Notice that for any edge   ∈ (), Algorithm 1 also guarantees ∆   = 0. Combining with (4.17) and (4.18) gives us the following equation This implies that (, ) is dual-optimal according to (4.7).Proof of Theorem 4.1.According to Proposition 4.3, (, ) is an optimal dual solution to (4.6).Moreover,  and  are obtained by additions and subtractions involving only the components of w.So (, ) is integral if w is integral, which completes our proof.
To summarize, MWBDRTP on trees can be reduced to a series of subproblems, and it thus can be solved using dynamic programming.An enhanced formulation incorporating some of the proposed new constraints has been proved to be TDI by showing that a integral dual optimal solution can always be obtained using a dual algorithm whenever the weights are integral.is a formulation for ℬ(, , c) if  is a cycle,   ≥ 2 and  = ∅.Hereafter we show that the system composed of ( 4

Dual algorithm
In order to present the dual algorithm, we introduce a notion called alternating edge set.An alternating edge set  (w) regarding the weight vector w ∈ R  is defined as such that it satisfies the following conditions The alternating edge set  (w) can then be written as Since in the remaining part of this paper, the weight vector w is always clear from the context, therefore  (or Thus, x  * and (, , ) are also optimal in this case.
If  = 0 and max{  :  ∈ } > 0, Algorithm 4 guarantees that for all (, ) ∈ ℳ with  (,) > 0, the following equation holds, Detailed proof of this equation can be found in [20].Consequently, one also has Therefore, x  * and (, , ) are always feasible and optimal.Finally, vectors ,  and  are obtained by additions and subtractions involving only the components of w.So (, , ) is integral if w is integral, which completes our proof.
Besides, in [20] it is shown that there exist upload-capacity inequalities and rooted matching-partition inequalities on trees and cycles with Chvátal-Gomory rank at least 2. This indicates that the characterization of ℬ(, , c) on trees and cycles cannot be trivially obtained as the first Chvátal closure of the polytope defined by (2.1)-(2.5).

Decomposition at the root
Consider a connected graph  = (, ) where  is an articulation node, such that  is a 1-sum of  1 = ( 1 ,  1 ) and  2 = ( 2 ,  2 ) at .Given a vector x in R  , let x  be the restriction of x to   ,  = 1, 2. Additionally, let the capacity vector on graph   be c  ∈ Z  , such that    =   for any  ∈   ,  = 1, 2. The following polytope and ℬ(, , c) can be proved to be identical in this case.Proof.It is straightforward to see that   (, , c) ∩ Z  = ℬ(, , c) ∩ Z  , or in other words, a bounded-degree rooted tree of  is composed of two bounded-degree rooted trees of  1 and  2 respectively.Assume that there exists a fractional extreme point x in   (, , c).Let (x) be the linear system of equations that defines x.Without loss of generality, assume that (x) contains || equations (whose coefficient matrix has full rank).
On the other hand, if the articulation node is not , this decomposition will not work as straightforwardly.Take the graph in Figure 1 as an example.The following inequality defines a facet of ℬ(, , c).Inequality (5.1) has variables associated with edges in both  1 and edges in  2 .Hence if one wants to decompose  into  1 and  2 , inequalities such as (5.1) should be included in addition to the simple combination of polytopes respecting  1 and  2 .Beside the inequalities introduced previously, there are more inequalities can be found to be facet-defining, which involves the factor of capacity.Given a graph as demonstrated in Figure 3A with   =   = 2 and capacity of any other nodes being sufficiently large, one has an inequality  1 +  2 −  1 −  2 ≤ 0, which is facet-defining.
Additionally, if the edge  3 is expanded into a path as in Figure 3B, one can also get a facet-defining inequality as the following one.
Nonetheless, it is found out that the following series of inequalities can be obtained in this case and are all facet-defining.It can be noticed that the four aforementioned inequalities only differ in the coefficients of  3 ,  4 , and  5 .Furthermore, they do not belong to any set of inequalities that have been introduced previously, and their graphical interpretation is yet to be revealed.Thus, the decomposition over an arbitrary articulation node is hitherto unlikely to work to the best of our knowledge.

Concluding remarks
In this paper, the polytope associated with MWBDRTP is studied.The dimension of the polytope is examined first.Several sets of valid inequalities and their facet-defining conditions are discussed.With two families of newly proposed facet-defining inequalities, the polytope is proved to be characterizable with a TDI system in each case on trees and cycles.Additionally, the decomposition of the polytope with respect to the articulation nodes is proved to be feasible if the articulation node is the root.
Besides the aspects examined in this paper, there are a few directions can be further explored for MWBDRTP.On the one hand, the application of MWBDRTP in the telecommunication field considers a packing of potentially more than one rooted trees.This problem is called the Maximum-Weight Bounded-Degree Rooted Tree Packing Problem (MWBDRTPP).Preliminarily, we have looked into the case of 2 rooted trees as the first step.The polyhedral structure turns out to be much more complicated.With a formulation also considering only edgeindexed variables, we have characterized some fractional extreme points in the case where 2 rooted trees are considered and the graph  is a star, which can be cut by the following constraint x 1 (()) + x 2 (()) − x 1 (()) − x 2 (()) ≤   for all  ∈  ⊆  ∖ {}, where the superscripts correspond to the index of the rooted trees.Nonetheless, considering a packing of 2 rooted trees, a polynomial-time combinatorial algorithm for MWBDRTPP on trees is proposed in the work with [19].
On the other hand, we have also done some computational testing on different formulations for ℬ(, , c), in order to see how the new inequalities, presented in this paper and some others introduced in [13], affect the performance of a branch-and-cut algorithm on graphs with different properties.Generally, these new inequalities are able to improve the performance of the branch-and-cut algorithm in terms of gap, number of solved instances and running time, although the improvements vary as the graph property changes (e.g., parse graphs vs. dense graphs).Detailed results and discussions can be found in [13] (and in [20] as well).

Theorem 4 .
1 can then be proved based on Proposition 4.3.

Figure 1 .
Figure 1.Counter example of decomposition involving two 2-connected components.

Figure 2 .
Figure 2. Counter example of decomposition involving rooted matching-partition inequalities.
Proof of Lemma 2.2. being connected guarantees that   exists for any edge  ∈  .Moreover for any two distinct edges  1 ,  2 ∈  , if | 1 | ≥ | 2 | then one trivially has  1 ̸ ∈  2 .Suppose that there exists a non-zero vector  ∈ R  such that Let  + = { ∈  :   ̸ = 0} and let   be an edge in  + such that |  | ≥ |  | for any  ∈  + .One therefore has   / ∈   for any  ∈  + ∖{  }.Consequently one deduces   = 0, a contradiction with   ∈  + .The vectors in the set {x  :  ∈  } thus are linearly independent and hence affinely independent.Proof.Let   = (  ,   ) be the connected component containing  of the subgraph [ ∖].By Assumption 1.1  is a stable set of  and therefore,   =  ∖  and   =  ∖ ().Given any  ∈   , let   be a shortest - path in [  ].Since   contains no nodes in ,   is a - path that satisfies the capacity constraints and contains as few edges as possible.According to Lemma 2.2, the vectors in the set   = {x  :  ∈   } are affinely independent.Each of these vectors also satisfies the capacity requirement for no unit-capacity nodes are involved in those paths.Consider any edge   =   ∈   () where   ∈  and then  ∈   .Let   denote an - path in [  ] and   =   ∪ {  }.All the inner nodes of   have capacity of value at least 2 for they belong to   .  then is a bounded-degree rooted tree of .Clearly the set  =   ∪ {x   :   ∈   ()} is composed of |  | + |  (  )| = || affinely independent vectors for x   is the only vector satisfying   = 1.
.8)4.1.2.Dual algorithm and TDI-nessFor any  ∈ , let   be the dual variable corresponding to inequality (4.1) and (4.4) associated with .For any  ∈  , let   be the dual variable corresponding to inequality (4.2) and (4.3) associated with .The dual linear program of (4.6) is min     + ∑︁     and   is the extremity of  the further away from .Note that the reason why we can only use two sets of dual variables ( and ) is that although we need to define a dual variable for each inequality, equations (4.1) and (4.4) have similar forms and are associated with edges, and (4.2) and (4.3) have similar forms and are associated with nodes.As a result, we can combine them into only two sets rather than four.