Minimum Color-Degree Perfect b -Matchings

Funding information This work was supported by the Freigeist-Fellowship of the Volkswagen Stiftung; and the German research council (DFG) Research Training Group 2236 UnRAVeL. Abstract The minimum color-degree perfect b-matching problem (Col-BM) is a new extension of the perfect b-matching problem to edgecolored graphs. The objective of Col-BM is to minimize the maximum number of differently colored edges in a perfect b-matching that are incident to the same node. We show that Col-BM is NPhard on bipartite graphs by a reduction from (3,B2)-Sat, and conclude that there exists no (2 − ε)-approximation algorithm unless P = NP. However, we identify a class of two-colored complete bipartite graphs on which we can solve Col-BM in polynomial time. Furthermore, we use dynamic programming to devise polynomialtime algorithms solving Col-BM with a fixed number of colors on series-parallel graphs and simple graphs with bounded treewidth.


| INTRODUCTION
Assignment problems are among the most famous combinatorial optimization problems. In its most basic form, the assignment problem consists of a set of agents A, a set of jobs B, and a set of agent-job pairs E ⊆ A × B that define which agent can perform which job [1]. The objective is to find a one-to-one assignment of jobs to agents. Graphtheoretically the assignment problem corresponds to the maximum (weighted) matching problem in a bipartite graph which is known to be polynomial-time solvable by the Hungarian method [2]. However, for many applications this original version of the assignment problem fails to capture all relevant requirements. Therefore, various more complex forms of the assignment problem are studied, e.g., the (capacitated) b-matching problem [1] or the restricted matching problem [3]. In the maximum b-matching problem, there is a specified b-value for every node which determines In this paper we study a new extension, the so-called minimum color-degree perfect b-matching problem (Col-BM), which we introduce via the following application; see Figure 1. Imagine that an airline aims to establish new flight connections using different types of aircraft. The appropriate type of aircraft is given for every connection of interest, and the number of operable connections at each airport is dictated by the takeoff and landing slots owned by the airline. As unused slots have to be returned permanently by policy so that they can be reassigned to other airlines [4], all available slots at all airports have to be utilized. However, operating different types of aircraft at the same airport decreases flexibility in crew scheduling, and increases the necessary space for spare-part storage. Therefore, the maximum number of different types of aircraft operated at any airport should be minimized.
In the setting above, the selection of appropriate flight connections corresponds to a perfect simple b-matching problem, which consists in finding an edge subset of a graph such that the vertices in the resulting subgraph have certain prespecified degrees. However, a classical b-matching neglects the diversity induced by different types of aircraft. We model the different types of aircraft by adding colors to the edges of the underlying graph. This leads to Col-BM, a b-matching extension on an edge-colored graph with the objective of minimizing the maximum number of differently colored edges incident to the same node.
Before providing a formal definition of Col-BM, we introduce some notation that is used in the paper. Let In this paper, we study the complexity of Col-BM on different graph classes. Our main contributions can be summarized as follows: • Col-BM is strongly N P-hard on two-colored bipartite graphs • There exists no (2 − ε)-approximation algorithm for Col-BM unless P = N P.
• Col-BM on two-colored complete bipartite graphs G = (V A ∪ V B , E ), with b (v ) = 1 for all v ∈ V A and b (v ) = 2 for all v ∈ V B can be solved in O ( |V | 2 ) time.
• Col-BM with a fixed number of q colors on series-parallel graphs can be solved in O ( |E | · max v ∈V b (v ) 4 ) time.
• Col-BM with a fixed number of q colors on simple graphs G = (V , E ) with treewidth tw(G ) < W can be solved With these results, we extend ongoing studies of matching problems on edge-colored graphs. Most contributions in this field incorporate restrictions depending on the edge coloring in order to reduce the space of feasible solutions.
The probably first problem of this kind is the rainbow (or multiple-choice) matching problem [6]: Given an edge-colored graph, find a maximum matching such that all edges have distinct colors. The rainbow matching problem is known to be N P-complete on bipartite graphs [7], and Le and Pfender [8] more recently proved that it is even A P X-complete on paths. Another problem of this kind is the blue-red matching problem (BRM): Given a blue-red-colored graph and w ∈ 0 , find a maximum matching which consists of at most w blue and at most w red edges. Nomikos et al. [9] devised an R N C 2 as well as an asymptotic 3 4 -approximation algorithm for BRM. The exact complexity of BRM is still open.
One of the earliest weighted matching problems considered on edge-colored graphs is the bounded color matching problem (BCM): Given an edge-colored graph with edge weights, find a maximum weighted matching such that the number of edges in each color satisfies a color-specific upper bound. As a generalization of rainbow matching, all complexity results of the former directly translate to BCM. A straight forward, greedy strategy leads to a 1 3 -approximation algorithm for BCM [10]. Moreover, several bi-criteria approximation algorithms for BCM, which are allowed to slightly violate the color constraints, are due to Mastrolilli and Stamoulis [10,11]. Recently, an extension of BCM that additionally incorporates edge costs was studied under the name budgeted colored matching problem [12]. Büsing and Comis [12] present pseudo-polynomial dynamic programs for the budgeted colored matching problem with a fixed number of colors on series-parallel graphs and trees.
The concept of incorporating the edge-coloring into the objective function of a matching problem is, to our knowledge, relatively new and only few problems of this type have been studied yet. One such problem that is closely related to Col-BM is the labeled maximum matching problem (LMM): Given an edge-colored graph, LMM asks for a maximum matching that uses the minimum number of different colors. Monnot [13] showed that LMM is A P X-complete on bipartite complete graphs and 2-approximable on 2-regular bipartite graphs. Subsequently, Carrabs et al. [14] presented alternative mathematical formulations and an exact branch-and-bound scheme for LMM. Another family of related problems are so-called reload cost problems. In reload cost problems, the edge colors symbolize different types of transport, and costs arise for every change of color at a node. The task is to find a specific subgraph for which the weighted sum of all color changes is minimal. As subgraphs have been considered, e.g., spanning trees [15], paths between two vertices [16], tour or flow problems [17]. For a detailed review of this kind of problems we refer to [18].
A weighted b-matching problem with an objective function incorporating the edge coloring is the diverse weighted b-matching problem (D-WBM). D-WBM can be considered as the counterpart of Col-BM: Given a weighted, edgecolored, bipartite graph, D-WBM asks for a b-matching satisfying upper and lower vertex degree bounds such that the weights of edges incident to the same node are evenly distributed among all colors. In [19], D-WBM is claimed to be N P-hard and diversification is ensured by minimizing a quadratic function that penalizes unbalanced weight-color distributions rather than adopting a Max-Min approach analogous to our Min-Max approach. For a more extensive review on general matching theory we refer to [10]. This paper is organized as follows. In Section 2, we prove that Col-BM is N P-hard in general. However, in Section 3, we identify a class of two-colored complete bipartite graphs for which Col-BM is solvable in polynomial time. Furthermore, we provide dynamic programs for solving Col-BM on series-parallel graphs (Section 4) and on simple graphs with bounded treewidth (Section 5) that run in polynomial time if the number of colors is fixed. We close this paper with a conclusion and an outline of future research (Section 6).

| COMPLEXITY
Concerning the complexity of Col-BM, we remark that if b (v ) = 1 for all v ∈ V (G ), Col-BM reduces to a simple, polynomial-time solvable perfect matching problem. In the following we show that in general the decision version of

Theorem 2. The decision version of Col-BM on two-colored bipartite graphs
Proof We reduce (3,B2)-Sat to the decision version of Col-BM. The problem (3,B2)-Sat is a strongly N P-complete

Layer 2
Layer 1 variables x 1 , . . . , x n and clauses C 1 , . . . , C m . We construct a corresponding Col-BM instance where G is composed of two layers; see Figure 2. Layer 1 models the correspondence between a perfect b-matching with color degree one and a satisfying truth assignment for an instance of (3,B2)-Sat. Layer 2 is an auxiliary, complete bipartite graph ensuring the existence of a perfect b-matching. In the following we refer to edges in E 1 as blue edges and to edges in E 2 as red edges. . . , 7 6 n } }, ensuring the existence of a perfect b-matching in G . Note that 7 3 n is integer as 3 divides n. If 7 3 n is even, we define W := ∅ and otherwise W := {w 0 }. We connect W with V and R via the following edges: a red-colored edge {v i , w i ,3 } and blue-colored edges {v i , w i ,1 }, {v i , w i ,2 } for each i ∈ {1, . . . , n }, as well as blue-colored edges {r , w } for all r ∈ R and w ∈ W . Finally, we set b (w ) = 1 for all w ∈ W and b (r ) = 2 for all r ∈ R . As a result, G is bipartite by construction  i , u j } is blue, then x i ∈ C j by construction. Hence, our choice x i = True verifies clause C j . Analogously, if {v i , u j } is red, then x i ∈ C j . Hence, our choice x i = False verifies clause C j . Consequently, x is a satisfying assignment for I.

Conversely
, let x be a satisfying truth assignment for I and M = ∅. We choose a verifying literal x i (x i ) for each clause To conclude our reduction, it suffices to extend M to Layer 2 without increasing f max G (M ). Therefore, we proceed for , which exists as G is a complete bipartite graph and, by construction, As the decision version of Col-BM is obviously in N P, as we can check the feasibility and color-degree of a given b-matching in O ( |V (G ) | · |E |) time, the problem's strong N P-completeness follows. perfect b-matching in G has color degree one or two. This directly implies the inapproximability of Col-BM.
Note that any b-matching in a two-colored graph has color degree at most two. Hence, every exact algorithm solving the perfect b-matching problem is a 2-approximation algorithm for Col-BM on two-colored graphs.
Remark that Col-BM on a two-colored bipartite graph corresponds to the task of partitioning G into monocromatic paths of length 3 whose end-nodes are exclusively in V A (spanning P 3 Partition). It is known that partitioning an uncolored graph into paths of length 3 (P 3 Partition) is N P-complete on bipartite graphs of maximum degree 3 [21]. However, to the best of our knowledge, no work has been published on monochromatic P 3 Partition problems in edge-colored graphs nor on spanning P 3 Partition problems in uncolored graphs. In the case that b (v ) = r , r ∈ , for all v ∈ V , Col-BM is closely related to the partitioning of graphs into monochromatic r -factors. A survey on partitioning problems of edge-colored graphs into monochromatic subgraphs can be found in [22].

| COMPLETE BIPARTITE GRAPHS
In the previous section, we have proven that Col-BM is N P-hard on two-colored bipartite graphs In this section, we additionally assume G to be complete bipartite and prove that in this case Col-BM is solvable in polynomial time by providing a constructive algorithm. For better lucidity we abbreviate the edge notation {v , w } as vw in this section. We utilize two characteristics of such graphs to classify those for which a perfect b-matching M with f max G (M ) = 1 exists. We begin by identifying a subgraph which is sufficient for the existence of a perfect b-matching M with Proof Let G be a graph that contains the subgraph G . We present an algorithm to construct a perfect b-matching M in G with f max G (M ) = 1. Therefore, let G be the subgraph defined above and initialize M = ∅. For a given M ⊆ E , Repeat the following two steps until all w ∈ V B \ V (G ) are satisfied. First, choose a node w ∈ V B \ V (G ) and three Hence, it suffices to prove that there always exists a perfect b-matching M in the induced We distinguish two cases based on the color of the edge ar ; see Figure 3(b): As not all instances contain the gadget G , we continue by exploiting the fact that in every perfect b-matching M in As a result, for every node v ∈ V B only node pairs that are connected to v by edges of the same color are potential matching partners.

Definition 5.
Let G be a q -colored graph. For v ∈ V (G ) and a color i ∈ {1, . . . , q }, we define the i -colored neighborhood

Remark that in a complete bipartite graph
we call G stable (color) partitioned; see Figure 4. We use the notion of a stable partitioning to determine whether a is even for all colors i ∈ {1, . . . , q } and w ∈ V B .
Proof Let {P 1 , . . . , P q } denote the unique partition of V A induced by the set of i -colored neighborhoods of r ∈ V B .
On the one hand, if |P i | is even for all i ∈ {1, . . . , q }, then we construct a perfect b-matching M with f max G (M ) = 1 by iteratively matching two unsatisfied nodes belonging to the same class P i to an unsatisfied node of V B .
On the other hand, if M is a perfect b-Matching with f max G (M ) = 1, then every P i is canonically partitioned by M into disjoint node pairs. Thus, |P i | has to be even for all i ∈ {1, . . . , q }.
We proceed by proving that every two-colored complete bipartite graph G = (V , E ) with |V | > 6 either fulfills the conditions of Lemma 4 or the conditions of Lemma 6. This leads to a complete characterization of two-colored complete bipartite graphs with more than six nodes and will be used to derive an algorithm for this graph class.
Then exactly one of the following is true.

1) G contains the gadget G defined in Lemma 4.
2) G is stable partitioned.
Proof Assume 1) holds. Then 2) is violated as r and s induce different partitions of {b, c, d }.
at least one of the following holds Remark that as |V (G ) | > 6 and |V A | = 2 |V B |, it directly follows that |V A | 6 and |V B | 3.
Without loss of generality assume that i) holds, as the argumentation is analogous in the case that ii) holds. The va- Therefore, bs, cr , cs and ds are of color one whereas br is of color two; see Figure 5. Consequently, ( {b, c, d , r , s }, {br , bs, cr , cs, ds }) represents a gadget as defined in Lemma 4. If N 1 (s) = 2 and ii) holds, then N 2 (s) 3 and the statement follows via symmetry. Therefore, assume that N 1 (s) = 2 and ii) is violated. Then either N 1 (r ) = 1 or N 2 (r ) = 1; see Figure 6. If and d ∈ N 2 (r ) \ {b, c }. Therefore, br , cr , cs and d r are of color two whereas bs is of color one. Consequently, ( {b, c, d , r , s }, {br , bs, cr , cs, d r }) represents a gadget as defined in Lemma 4; see Figure 6(a). If We conclude, if 2) is violated, then 1) holds.
Remark that the condition imposed on the size of the graph in Lemma 7 is tight.

Proposition 8.
There exists a graph G with |V (G ) | = 6 that is neither stable partitioned nor does it contain the gadget G .
Proof The graph neither is stable partitioned nor contains the gadget G ; see Figure 7.
The results from Lemma 7 imply that, on a two-colored complete bipartite graph Col-BM can be reduced to identifying the gadget G as subgraph, or determining that none exists; see Algorithm 1: We first check whether G is stable partitioned. If this is the case, G does not contain the gadget G and we can determine the minimum color degree of a perfect b-matching in G by checking the cardinalities of the elements of the unique partition of V A : if all cardinalities are even, the minimum color degree of a perfect b-matching in G is one otherwise it is two. In the other case, G contains the gadget G and, hence, the minimum color degree of a perfect b-matching in G is one.

| SERIES-PARALLEL GRAPHS
In this section, we consider Col-BM on series-parallel (SP)-graphs. We show that, in case of a fixed number of colors, Col-BM can be solved in polynomial time on SP-graphs by dynamic programming. Subsequently, we extend our dynamic program to solve Col-BM on trees.
We start with a formal definition of SP-graphs based on the one given in [23].

Definition 10.
A (2-terminal) SP-graph with two distinguished nodes σ and τ , called source and sink, is defined as follows.
2) A graph, constructed by a finite number of the following operations, is SP.
The series and parallel composition are visualized exemplarily in Figure 8(a).
Every SP-graph G can be associated with a decomposition tree T = T (G ), which is a rooted, binary tree whose nodes correspond to the subgraphs of G appearing in the recursive construction; see Figure 8(b). The leaves of the decomposition tree correspond to edges in G . The inner nodes of the decomposition tree are of two different types: an S -node corresponds to the series-composition of the graphs associated with its child nodes and, analogously, a Pnode corresponds to the parallel composition of its child nodes. We denote the root of T with r and it corresponds to G itself by construction.
Let G = (V , E ) be an SP-graph with edge coloring E 1 ∪ . . . ∪E q = E , and b : V → 0 a mapping. It is known that a decomposition tree can be computed in linear time for an SP-graph [24]. Thus, let T be a decomposition tree for G .
For t ∈ V (T ), let G t denote the subgraph of G with source σ t and sink τ t corresponding to t . We propose a dynamic program to solve Col-BM on SP-graphs using the corresponding decomposition trees.
First, we introduce a set of labels for every t ∈ V (T ). The parameters α and β define new, smaller b-values for σ t and τ t , whereas the color-subsets F σ , F τ define the prespecified set of colors for edges incident to σ t and τ t .
Before we specify our dynamic program, we introduce some more notation. For a node t ∈ V (T ) and a label x = For a node t ∈ V (T ) and a label x ∈ L t , we call the optimal solution value of the (t , x )-restricted Col-BM the cost c t (x ) of x at t . Thus, for all perfect b-matchings M * in G with minimum color degree it holds that for the root r of T . Our dynamic program solving the Col-BM on SP-graphs exploits the structure of decomposition trees and recursively computes label costs bottom-up. To that end, we consider the three types of nodes in the decomposition tree of G starting with the initialization in leaves. For the two remaining types of tree nodes, label costs can be derived recursively from the label costs of child nodes.
We begin by considering S -nodes, which correspond to the series composition of the graphs associated with its child nodes. As a result of this interrelation, every restricted matching at an S -nodes can be decomposed into two restricted matchings at its child nodes. By minimizing over all feasible combinations of restricted matchings at the child nodes, we get the following.

Lemma 12.
Let t ∈ V (T ) be an S -node in T with child nodes and u. Then the cost of x t = α t , F t σ , β t , F t τ ∈ L t at t can be computed as Proof If t ∈ V (T ) is an S -node with child nodes and u, by definition σ t = σ , τ t = τ u , and τ = σ u =: y . Let Furthermore, fork := δ M (y ) ,F τ := col M (y ), andF u σ := col M u (y ) it holds that M is an , α t , F t σ ,k ,F τrestricted matching in G while M u is a u, b (y ) −k ,F u σ , β t , F t τ -restricted matching in G u . Thus by definition, Conversely, let Moreover, let M ⊆ E (G ) be an optimal solution to the , α t , F t σ , k * , F * τ -restricted Col-BM on G and M u ⊆ E (G u ) be an optimal solution to the u, b (y ) − k * , F u * σ , β t , F t τ -restricted Col-BM on G u . We define the matching M t := M ∪ M u in G t . By construction, M t is (t , x t )-restricted, col M (τ) = F * τ , and col M u (σ) = F u * σ . Thus, To conclude the computation of label costs, we consider P -nodes. Recall, that P -nodes correspond to the parallel composition of the graphs associated with its child nodes. Thus, we can again compute the cost of labels by minimizing over all feasible combinations of restricted matchings at the child nodes.
Lemma 13. Let t ∈ V (T ) be a P -node in T with child nodes and u. Then the cost of x t = α t , F t σ , β t , F t τ ∈ L t at t can be computed as Proof If t ∈ V (T ) is a P -node with child nodes and u, by definition σ = σ u = σ t and τ = τ u = τ t . Let (2) For the choice ofk : Conversely, let Moreover, let M ⊆ E (G ) be an optimal solution to the , k * , F * σ , m * , F * τ -restricted Col-BM on G and M u ⊆ E (G u ) be an optimal solution to the u, α t − k * , F u * σ , β t − m * , F u * τ -restricted Col-BM on G u . We define the matching A perfect b-matching M * in G of minimum color degree can be obtained by backtracking the chosen minima in the steps of the dynamic program.
Next, we consider the runtime of our dynamic program. For better lucidity, let B = max v ∈V b (v ).

| GRAPHS WITH BOUNDED TREEWIDTH
We proceed by considering Col-BM on graphs with bounded treewidth, which is a more general graph class that includes SP-graphs. Using dynamic programming, we show that Col-BM on graphs with bounded treewidth is polynomial-time solvable for a fixed number of colors. Before we present the details of our algorithm, we introduce the concept of tree decompositions, followed by the definition of a graph's treewidth according to Robertson and Seymour [26]. 1) Every vertex of G is contained in at least one bag, i.e., t ∈V (T ) X t = V .

2)
For each edge {v , w } ∈ E there exists a node t ∈ V (T ) such that v , w ∈ X t .
3) For all vertices t , , u ∈ V (T ) such that lies on the unique path between t and u in T , it holds that X t ∩X u ⊆ X .
The width of a tree decomposition of graph G is defined as the cardinality of its largest bag minus one, i.e., For better lucidity of dynamic programs, so-called nice tree decompositions were defined as a subclass of all tree decompositions [27].

Definition 17.
A tree decomposition (T , X) of a graph G = (V , E ) is called nice if T is a rooted tree and all nodes t ∈ V (T ) can be categorized into four groups:

1)
Leaves t ∈ V (T ) have no child nodes and their bag contains exactly one vertex v ∈ V , i.e., X t = {v }.
2) Introduce nodes t ∈ V (T ) have exactly one child node ∈ V (T ) such that X X t and X t \ X = {w } for some w ∈ V .
3) Forget nodes t ∈ V (T ) have exactly one child node ∈ V (T ) such that X t X and X \ X t = {w } for some w ∈ V .

4)
Join nodes t ∈ V (T ) have exactly two child nodes , u ∈ V (T ) such that X t = X = X u .
For any graph G = (V , E ) with bounded treewidth tw(G ) < W , a nice tree decomposition (T , X) with O ( |V |) nodes and tw(G, (T , X)) < W can be computed in linear time [28,29]. An exemplary graph with corresponding nice tree decomposition is visualized in Figure 9.
Our dynamic program for solving the Col-BM on graphs with bounded treewidth exploits the structure of nice tree decompositions and recursively computes label costs bottom-up. Let G = (V , E ) be a graph with bounded treewidth tw(G ) < W ∈ , E 1 ∪ . . . ∪ E q = E be an edge coloring of G , andc : E → {1, . . . , q } the corresponding color function.
Further, let (T , X) be a nice tree decomposition of G such that tw(G, (T , X)) < W . Without loss of generality, we assume that the bag X r , corresponding to the root r of T , contains exactly one vertex. Should (T , X) violate this assumption, we simply add a sequence of forget nodes to r and redefine T 's root.
For a tree node t ∈ V (T ) we denote the set of edges of G induced by its bag X t with E [X t ] and the subgraph of G induced by the vertices in the bags of the subtree of T rooted in t with G t . As before, for a vertex v ∈ V and a subset of edges M ⊆ E , we denote the set of colors in δ M (v ) by col M (v ). Finally, for all mappings f : A → B, we abbreviate f a := f (a) for a ∈ A for ease of notation.
We use labels of the form Finally, the mapping β : X t → 0 defines the required degree of each vertex v ∈ X t with respect to matching edges We formalize the auxiliary problem xCol-BM(t ,x ) as follows: Every b-matching in G t satisfying the constraints (3) - (7) is called (t , x )-feasible. We define the cost c t (x ) of label x ∈ L t at tree node t as the optimal solution value to xCol-BM(t ,x ). If xCol-BM(t ,x ) is infeasible, we call x invalid and we set c t (x ) = +∞. All remaining labels are called valid and we calculate their cost recursively. To that end, we consider the four types of nodes in the nice tree decomposition (T , X) of G starting with the initialization in leaves.

Lemma 18.
Let t ∈ V (T ) be a leaf with X t = {v } for some v ∈ V . Then the cost of a valid label x = (m , F , β ) ∈ L t at t can be computed as Proof As t is a leaf, E [X t ] = ∅ and G t consists of the isolated vertex v ∈ X t . All valid labels x ∈ L t have the . . , q }, and β v = 0. The only (t , x )-feasible matching in G t is M := ∅ and thus, For the three remaining types of tree nodes, label costs can be derived recursively from the label costs of child nodes.
We begin by considering introduce nodes.

Lemma 19.
Let t ∈ V (T ) be an introduce node with unique child node ∈ V (T ), and let w ∈ V be the introduced vertex, i.e., X t \ X = {w }; see Figure 10(a). Given a valid label x t = (m t , F t , β t ) ∈ L t , we define the label x = (m , F , β ) ∈ L via m e := m t e for all e ∈ E [X ], F v := F t v for all v ∈ X , and β v := β t v for all v ∈ X . Then the cost of x t at t can be computed as being an introduce node with child node . (b) Visualization of G t for t ∈ V (T ) being a join node with child nodes , u.
Proof We begin by showing c t (x t ) max{c (x ), F t w }. Let M t be an optimal solution to xCol-BM(t , x t ). For the vertex w ∈ V introduced by node t ∈ V (T ), it holds that col M t (x t , w ) = F t w as w ∈ X t . Hence, and hence conditions (5) -(7) are satisfied. Therefore, M is ( , x )-feasible.
Conversely, we show that c t (x t ) max{c (x ), F t w }. Let M be an optimal solution to xCol-BM( , x ). We define the matching M t := M ∪ {e ∈ U | m t e = 1} and show that M t is (t , x t )-feasible in order to bound c t (x t ) from above.  (4), For the introduced vertex w ∈ X t , constraints (6) and (7) hold by the validity of x t . For v ∈ X t \ {w } = X , equation (6) holds as By the validity of x t , condition (7) holds for v ∈ X t \ {w } as Finally, equations (3) and (6) in combination with the validity of x t imply that inequalities (5) hold for all v ∈ V (G t ).
Therefore, M t is (t , x t )-feasible.
We conclude c t (x t ) = max{c (x ), F t w }.
Next, we consider the computation of label costs for forget nodes.
Lemma 20. Let t ∈ V (T ) be a forget node with unique child node ∈ V (T ). Let w ∈ V be the forgotten vertex, i.e., {w } = X \ X t , and denote its incident edges with respect to G [X ] by U : Then the cost of x t at t can be computed as To that end, note that G t = G and let M t be an optimal solution to xCol-BM(t ,x t ). We define a label x = (m , F , β ) ∈ L as follows: e ∩ M t for e ∈ U , (3) are satisfied for all e ∈ U , whereas e ∩ M t = m t e = m e for e ∈ E [X t ] since M t is (t , x t )-feasible. As V (G ) \ X ⊆ V (G t ) \ X t , constraints (4) and (5) hold by the x t -feasibility of M t . Concerning equalities (6), for any whereas for the forgotten vertex w it holds that Finally, as M t is (t , x t )-feasible, constraints (7) are satisfied by our definition of x and thus M t is ( , x )-feasible.
For the forgotten vertex w , F w = col M t (w ) by our definition of x . Thus, we conclude that and it follows that Conversely, for all labels x ∈ L (x t ), xCol-BM(t ,x t ) is a relaxation of xCol-BM( ,x ), and therefore To conclude the computation of label costs, we consider join nodes.

Lemma 21.
Let t ∈ V (T ) be a join node with child nodes and u. Given a valid label x t = (m t , F t , β t ) ∈ L t , we define the set L ,u (x t ) L × L u of pairs of labels at and u via L ,u (x t ) := (m ,F ,β ),(m u ,F u ,β u ) ∈ L ×L u m e =m u e =m t e e ∈ E [X t ], Then the cost of x t at t can be computed as Proof Recall that for join nodes X t = X = X u and (V (G ) \ X ) ∩ (V (G u ) \ X u ) = ∅; see Figure 10(b).
We begin by showing c t (x t ) be an optimal solution to xCol-BM(t ,x t ), and M := M t ∩ E (G ) and M u := M t ∩ E (G u ) the restrictions of M t to the subgraphs G and G u , respectively. We define labels x := (m t , F t , β ) ∈ L and x u := (m t , F t , β u ) ∈ L u such that and β u v = δ M u (v ) \ E [X t ] for all v ∈ X t . The x t -feasibility of M t implies for all vertices v ∈ X t that By construction, the matchings M and M u are feasible for xCol-BM( ,x ) and xCol-BM(u,x u ), respectively. Moreover, To that end, consider a pair of labels (x , x u ) ∈ L ,u (x t ), and let M and M u be optimal solutions to xCol-BM( ,x ) and xCol-BM(u,x u ), respectively. We define the proving that equations (4) hold. Concerning constraints (6), for which, in combination with the validity of x t , directly implies that inequalities (5) are satisfied. Finally, for all v ∈ X t it proving the validity of constraints (7). Therefore, M t is (t , x t )-feasible.
and therefore, it holds in particular that We conclude c t (x t ) = min (x ,x u ) ∈L ,u (x t ) max{c (x ), c u (x u ) }.
Finally, we show how the optimal solution value to Col-BM on G is obtained from the computed label costs. Conversely, we show f max G (M * ) min x ∈L * c r x . Let x r ∈ L * and M r be an optimal solution to xCol-BM(r , x r ). We note that equations (4) and (6)  Concerning the algorithm's runtime, recall that a nice tree decomposition (T , X) of G with |V (T ) | ∈ O ( |V |) nodes can be computed in linear time. For each t ∈ V (T ), the number of labels L t we have to consider at t is in The computation of label costs for leaves and introduce nodes can be done in O (1) time.
For labels x ∈ L t at forget node t ∈ V (T ) with child node , we have to compare the label costs of L (x ) = 2 |U | 2 q labels. For simple graphs, |U | |W | and thus, the computation of label costs for forget nodes is in O (2 W 2 q ) time.
Consequently, the computation of label costs for join nodes is in O (B W ) time.
In conclusion the computation of label costs can be performed in O (max{2 W +q , B W }) time. This results in a total runtime in O ( |V | · 2 W 2 +W (q −1) · B W · max{2 W +q , B W }).

Corollary 24.
Col-BM on simple graphs G = (V , E ) with bounded treewidth tw(G ) < W is FPT with respect to the number of colors q , the width bound W , and the maximum b-value B.
We note that for all Col-BM instances B |E | and thus, for fixed q and W our dynamic program runs in polynomial time (O ( |V | · B 2W ) time) . For trees, which are simple graphs with treewidth 1, the runtime obtained from Theorem 23 coincides with the one from Corollary 15.
As soon as we drop the width bound W , we obtain Col-BM on general graphs with a fixed number of colors which is strongly-N P hard by Theorem 2, even for B = 2. The complexity of Col-BM on simple graphs G = (V , E ) with bounded treewidth tw(G ) < W and an arbitrary number of colors q remains open.

| CONCLUSIONS
In this paper we introduce the minimum color-degree perfect b-matching problem and prove its strong N P-hardness as well as its (2 − ε)-inapproximability on bipartite graphs with two colors. However, we identify a class of twocolored complete bipartite graphs on which we can solve Col-BM in quadratic time and propose polynomial-time dynamic programs solving Col-BM with a fixed number of colors on series-parallel graphs and simple graphs with bounded treewidth.
Future work includes generalizing the results for complete bipartite graphs to more colors as well as to more general b-values. Moreover, we will investigate the complexity of Col-BM on series-parallel graphs and graphs of bounded treewidth when the number of colors is not fixed. Furthermore, we plan to examine how special structures in the edge coloring can be exploited. Finally, we intend to devise general exact algorithms and heuristics for Col-BM by exploitation of structures in the underlying polytope.