Approximating Multistage Matching Problems

In multistage perfect matching problems, we are given a sequence of graphs on the same vertex set and are asked to find a sequence of perfect matchings, corresponding to the sequence of graphs, such that consecutive matchings are as similar as possible. More precisely, we aim to maximize the intersections, or minimize the unions between consecutive matchings. We show that these problems are NP-hard even in very restricted scenarios. As our main contribution, we present the first non-trivial approximation algorithms for these problems: On the one hand, we devise a tight approximation on graph sequences of length two (2-stage graphs). On the other hand, we propose several general methods to deduce multistage approximations from blackbox approximations on 2-stage graphs.


Introduction
The study of graphs that evolve over time emerges naturally in several applications. As such, it is a well-known subject in graph theory [1-11, 13, 14, 16-18, 21, 24]. While there are many possible approaches to model these problems (see the discussion of related work), the paradigm of multistage graphs has attracted quite some attention in recent years [2-5, 10, 14, 16, 17]. In this setting, we are given a sequence of separate, but related graphs (stages). A typical goal is to find a sequence of solutions for each individual graph such that the change in the solutions between consecutive graphs is minimized. Since multistage graph problems usually turn out to be NP-hard, one B Niklas Troost niklas.troost@uos.de Markus Chimani markus.chimani@uos.de Tilo Wiedera tilo.wiedera@uos.de 1 Theoretical Computer Science, Osnabrück University, Osnabrück, Germany often resorts to FPT-or approximation algorithms. To the best of our knowledge, all approximation results in this setting discuss combined objective functions that reflect a trade-off between the quality of each individual solution and the cost of the change over time (see, e.g., [2,16]). However, this is a drawback if one requires each stage's solution to attain a certain quality guarantee (e.g., optimality). Trying to ensure this by adjusting the trade-off weights in the above approximation algorithms leads to approximation ratios that no longer effectively bound the cost of change. Here, we discuss a multistage graph problem where each individual solution is necessarily optimal, but we can still obtain an approximation ratio on the cost of the change over time.
A classical example are multistage matching problems, i.e., natural multistage generalizations of traditional matching problems (e.g., perfect matching, maximum weight matching, etc.). This is particularly interesting as optimality for a single stage would be obtainable in polynomial time, but all known multistage variants are NP-hard already for two stages. There are several known approximation algorithms for multistage matching problems [2]; however, they all follow the trade-off paradigm.
In this paper, we are concerned with maintaining a perfect matching on a multistage graph, such that the changes between consecutive matchings are minimized. After showcasing the complexities of our problems (Sect. 2), we will devise efficient approximation algorithms (Sect. 3).

Definitions and Preliminaries
Let G = (V , E) be an undirected graph. For a set W ⊆ V of vertices, let δ(W ) := uv ∈ E | u ∈ W , v ∈ V \W denote the set of its cut edges. Let μ := max i∈[τ −1] |E i ∩ E i+1 | denote the maximum number of edges that are common between two adjacent stages. Let All problems considered in this paper (MIM, MUM, Min-MPM, Max-MPM; see below) are of the following form: Given a temporal graph G, we ask for a multistage perfect matching M optimizing some objective function. In their respective decision variants, the input furthermore consists of some value κ and we ask whether there is an M with objective value at most (minimization problems) or at least (maximization problems) κ. Definition 1 (MIM and τ -IM) Given a temporal graph G, the multistage intersection matching problem (MIM) asks for a multistage perfect matching M of G with maximum profit p(M) := i∈[τ −1] |M i ∩ M i+1 |. For fixed τ , we denote the problem by τ -IM.
We also consider the natural inverse objective, i.e., minimizing the unions. While the problems differ in the precise objective function, an optimal solution of MIM is optimal for MUM as well, and vice versa.
For fixed τ , we denote the problem by τ -UM.
Consider either MIM or MUM. Given a temporal graph G, we denote with opt the optimal solution value and with apx the objective value achieved by a given algorithm with input G. The approximation ratio of an approximation algorithm for MIM (MUM) is the infimum (supremum, respectively) of apx/opt over all instances.

Related Work
The classical dynamic graph setting often considers small modifications, e.g., single edge insertions/deletions [13,24]. There, one is given a graph with a sequence of modifications and asked for a feasible solution after each modification. A natural approach to tackle matchings in such graphs is to make local changes to the previous solutions [7][8][9]23].
A more general way of modeling changes is that of temporal graphs, introduced by Kempe et al. [18] and used herein. Typically, each vertex and edge is assigned a set of time intervals that specify when it is present. This allows an arbitrary number of changes to occur at the same time. Algorithms for this setting usually require a more global perspective and many approaches do not rely solely on local changes. In fact, many temporal (matching) problems turn out to be hard, even w.r.t. approximation and fixed-parameter-tractability [1,6,11,20,21].
One particular flavor of temporal graph problems is concerned with obtaining a sequence of solutions-one for each stage-while optimizing a global quantity. These problems are often referred to as multistage problems and gained much attention in recent years [2-5, 14, 16, 17], including in the realm of matchings: e.g., the authors of [17] show W[1]-hardness for finding the largest edge set that induces a matching in each stage.
In the literature we find the problem Max-MPM, where the graph is augmented with time-dependent edge weights, and we want to maximize the value of each individual perfect matching (subject to the given edge costs) plus the total profit [2]. MIM is the special case where all edge costs are zero, i.e., we only care about the multistage properties of the solution, as long as each stage is perfectly matched. There is also the inverse optimization problem Min-MPM, where we minimize the value of each perfect matching plus the number of matching edges newly introduced in each stage. We have APX-hardness for Max-MPM and Min-MPM [2,16] (for Min-MPM one may assume a complete graph at each stage, possibly including edges of infinite weight). The latter remains APX-hard even for spanning 2-stage graphs with bipartite union graph and no edge weights (i.e., we only minimize the number of edge swaps) [2]. For uniform edge weights 0, the objective of Min-MPM is to minimize i∈[τ −1] |M i+1 \M i |; similar but slightly different to MUM (equal up to additive i∈[τ −1] n i /2). For Min-MPM on metric spanning 2-or 3-stage graphs, the authors of [2] show 3-approximations. They also propose a (1/2)-approximation for Max-MPM on spanning temporal graphs with any number of stages, which is unfortunately wrong (see "Appendix A" for a detailed discussion).
When restricting Max-MPM and Min-MPM to uniform edge weights 0, optimal solutions for MIM, MUM, Max-MPM, and Min-MPM are identical; thus MIM and MUM are NP-hard. However, the APX-hardness of Min-MPM does not imply APX-hardness of MUM as the objective functions slightly differ. Furthermore, the APX-hardness reduction to Max-MPM inherently requires non-uniform edge weights and does not translate to MIM. To the best of our knowledge, there are no non-trivial approximation algorithms for any of these problems on more than three stages.

Our Contribution
We start with showing in Sect. 2 that the problems are NP-hard even in much more restricted scenarios than previously known, and that (a lower bound for) the integrality gap of the natural linear program for 2-IM is close to the approximation ratio we will subsequently devise. This hints that stronger approximation ratios may be hard to obtain, at least using LP techniques.
As our main contribution, we propose several approximation algorithms for the multistage problems MIM and MUM, as well as for their stage-restricted variants, see Fig. 1. In particular, in Sect. 3.1, we show a (1/ √ 2μ)-approximation for 2-IM and that this analysis is tight. Then, in Sect. 3.2, we show that any approximation of 2-IM can be used to derive two different approximation algorithms for MIM, whose approximation ratios are a priori incomparable. In Sect. 3.3, we further show how to use all these algorithms to approximate MUM (and 2-UM). We also observe that it is infeasible to use an arbitrary MUM algorithm to approximate MIM. In particular, we propose the seemingly first approximation algorithms for MIM and MUM on arbitrarily many stages. We stress that our goal is to always guarantee a perfect matching in each stage; the approximation ratio deals purely with optimizing the transition costs. Recall that approximation algorithms optimizing a weighted sum between intra-and interstage costs cannot guarantee such solutions in general.

Preprocessing and Observations
Given a graph G = (V , E), a single edge e is allowed if there exists a perfect matching M in G with e ∈ M and forbidden otherwise. A graph is matching-covered if Fig. 1 Relations of our approximation results. An arc from problem A to B labeled f (α) denotes the existence of an f (α)-approximation for B, given an α-approximation for A. In Corollary 16, α has to be constant. In Corollary 15, α(·) is a function of μ. The ratio of 2-IM is by Theorem 8; combining this with Theorem 17 yields the ratio for 2-UM all its edges are allowed (see [19] for a concise characterization of matching-covered graphs). Forbidden edges can easily be found in polynomial time; see e.g. [22] for an efficient algorithm. A simple preprocessing for MIM and MUM is to remove the forbidden edges in each stage, as they will never be part of a multistage matching. Thereby, we obtain an equivalent reduced temporal graph, i.e., a temporal graph whose stages are matching-covered. If any stage in the reduced temporal graph contains no edges (but vertices), the instance is infeasible. In the following, we thus assume w.l.o.g. that the given temporal graph is reduced and feasible, i.e., in each stage there exists some perfect matching.

Observation 3 Let G be a reduced 2-stage graph.
For any e ∈ E ∩ , there is a perfect matching in each stage that includes e. Thus, there is a multistage perfect matching with profit at least 1 if E ∩ = ∅.

Observation 4 For any multistage perfect matching
Thus, computing any multistage perfect matching is an immediate 2-approximation for MUM.

Setting the Ground
Before we present our main contribution, the approximation algorithms, we motivate the intrinsic complexities of the considered problems. On the one hand, we show that the problem is already hard in very restricted cases. On the other hand, we show that natural linear programming methods cannot yield a constant-factor approximation for 2-IM.

NP-Hardness
While it is known that 2-IM is NP-hard in general, we show that 2-IM is already NPhard in the seemingly simple case where each vertex has only degree 2 in both stages. It immediately follows that the decision variants of MIM, 2-UM, MUM, Min-MPM, and Max-MPM remain NP-hard as well, even if restricted to this set of temporal graphs.

Theorem 6 Deciding 2-IM is NP-hard on spanning temporal graphs whose union graph is bipartite, even if both stages consist only of disjoint even cycles and E ∩ is a collection of disjoint 2-paths.
Proof We will perform a reduction from MaxCut [15] to 2-IM. In MaxCut, one is given an undirected graph G = (V , E), a natural number k and the question is to decide whether there is an S ⊆ V such that |δ(S)| ≥ k. In the first stage, we will construct an even cycle for each vertex and each edge of the original graph and in the second stage we will create an even cycle for each incidence between an edge and a vertex (see Fig. 2). A perfect matching in the first stage will correspond to a vertex selection and a perfect matching in the second stage will allow us to count the edges that are incident to exactly one selected vertex.
Given an instance I := (G = (V , E), k) of MaxCut, we construct an instance J := (G, κ) of 2-IM. Set κ := 3|E| + k. We start with an empty 2-stage graph Mark one edge of each X e v as x e v and one edge of each Y e v as y e v . We will refer to the endpoint of . In G 1 , for each e = vw ∈ E, we create a 6-cycle through Y e v and Y e w by adding an edge between the marked endpoint of one path and the unmarked endpoint of the other, and vice versa. Furthermore, for each v ∈ V , we create a cycle of length 4|δ(v)| through the paths X e v for e ∈ δ(v) as follows: For each edge e ∈ δ(v) and its successor edge f , according to some arbitrary cyclic order of δ(v), connect the marked endpoint of X e v to the unmarked endpoint of X f v by a 2-path with a new inner vertex. In G 2 , for each (v, e) ∈ I , we generate a 6-cycle through X e v and Y e v by adding an edge between the marked and an edge between the unmarked endpoints of the 2-paths, respectively. G 1 consists of |V | + |E| and G 2 of 2|E| disjoint even cycles, thus G is reduced.

Claim J is a yes-instance if and only if I is a yes-instance.
Proof of Claim Since both stages of G consist only of pairwise disjoint even cycles and there are only two perfect matchings in an even cycle, a perfect matching in a stage is determined by choosing one edge in each cycle. For e = vw ∈ E, let Observe that for any multistage perfect matching Consider some edge e = vw ∈ E. The intersection M 1 ∩ M 2 contains two edges of X e . If e ∈ δ(S), it also contains two edges of Y e , one marked and one unmarked. The cycles of length 4|δ(v)| may have introduced an even number of vertices W ⊆ V that are isolated in G 2 . To make G spanning, we add to E 2 an even cycle on W . This neither interferes with E ∩ nor the profit p, since W is an independent set in the first stage G 1 .

Linear Programming Gap
Linear Programs (LPs)-as relaxations of integer linear programs (ILPs)-are often used to provide dual bounds in the approximation context. Here, we consider the natural LP-formulation of 2-IM and show that the integrality gap (i.e., the ratio between the optimal objective value of the ILP and the optimal objective value of its relaxation) is at least √ μ, even already for spanning instances with a bipartite union graph. Up to a small constant factor, this equals the (inverse) approximation ratio guaranteed by Algorithm 1, which we will propose in Sect. 3. This serves as a hint that overcoming the approximation dependency √ μ for 2-IM may be hard. In the context of classical (perfect) matchings, the standard ILP formulation and its LP-relaxation describe the very same feasible points (called matching polytope), which is the corner stone of the problem being solvable in polynomial time [19]. Given a 2-stage graph G = (V , E 1 , E 2 ), the natural LP-formulation for 2-IM starts with the product of two distinct such perfect matching polytopes. Let δ (v) denote all edges incident to vertex v in G , and let (M 1 , M 2 ) be a 2-stage perfect matching in G. For each ∈ [2], we model M via the standard matching polytope: For each e ∈ E there is an indicator variable x e that is 1 if and only if e ∈ M . The constraints (1a) below suffice for bipartite graphs; for general graphs one also considers the blossom constraints (1b). These ensure that for any odd-sized vertex set W , at most |W |/2 edges between those vertices can partake in a matching (see [19] for details). Additionally to these standard descriptions, for each e ∈ E ∩ we use a variable z e that is 1 if and only if e ∈ M 1 ∩ M 2 . We want to maximize p(M 1 , M 2 ) = e∈E ∩ z e , such that: Thereby, constraints (1c), together with the fact that we maximize all z-values, ensure that z e = min{x 1 e , x 2 e } in any optimal solution. Theorem 7 The natural LP for 2-IM has at least an integrality gap of √ μ, independent of the number μ of edges in the intersection.

Proof
We construct a family of 2-IM instances, each with bipartite union graph, parameterized by some parameter k. Each instance is reduced and has a maximum profit of 1, but its LP relaxation has objective value at least k + 1 = √ μ.
Fix some k ≥ 3. We construct G := G(k) = (V , E 1 , E 2 ) as follows (see Fig. 3 for a visualization with k = 3). [k] . Let E ∩ := {a i, j b i, j } i, j∈ k , i.e., the intersection contains precisely the natural pairings of the a and b vertices. We call these the shared edges. In E 1 , we additionally add edges {b i−1, j a i, j } i∈[k], j∈ k . Similarly, we add edges {b i, j−1 a i, j } i∈ k , j∈[k] to E 2 . Now, both stages consist of k + 1 disjoint paths of length 2k + 1 which are "interwoven" between the stages such that (i) every second edge in each path is shared (starting with the first), and (ii) any path in G 1 has exactly one edge in common with every path in G 2 . Let P i , ∈ [2], i ∈ k , denote those paths in their natural indexing. We make each stage connected by joining every pair of "neighboring" paths, each together with a c and a d vertex. More precisely, we add edges {c j a 0, j−1 , c j a 0, j , b [k] to E 2 ; the indexing function ϕ(i) := k − i + 1 ensures that these new edges are not common to both stages. (In fact, if we would not care for a spanning G, we could simply use "new" vertices instead of reusing c, d in G 2 .) This finishes the construction, and since G contains no forbidden edges, it is reduced. Since the inner vertices of any path P 1 i have degree 2 in G 1 , any perfect matching in G 1 either contains all or none of the path's shared edges. Assume some shared edge a 0, j b 0, j is in a perfect matching in G 1 . Let C be the path a 0,0 c 1 a 0,1 c 2 . . . c k a 0,k in G 1 . Recall that all c-vertices have degree 2 in G 1 . Since a 0, j is matched outside of C, all other a 0, j , j = j, have to be matched with these c-vertices. Thus, P 1 j is the only path that contributes shared edges to the matching. Conversely, since C contains one less c-vertex than a-vertices, any perfect matching in G 1 has to have at least (and thus exactly) one such path. As the analogous statement holds for G 2 and by the interweaving property (ii) above, any multistage perfect matching contains exactly one shared edge.
However, we construct a feasible fractional solution with objective value √ μ.
Let λ := 1/(k + 1). We set the x-and z-variables of all shared edges to λ, satisfying all constraints (1c). This uniquely determines all other variable assignments, in order to satisfy (1a): Since the inner vertices of each P i have degree 2 in G , the non-shared edges in these path have to be set to 1 − λ. Again consider path C: Each a-vertex in C has an incident shared edge that contributes λ to the sum in the vertex' constraint (1a); there are no other edges incident to C. Thus, we have to set x 1 c j a 0, j−1 = 1 − jλ and x 1 c j a 0, j = jλ such that, for each vertex in C, its incident variable values sum to 1. The analogous statements holds for the corresponding path through d-vertices in G 1 , and the analogous paths in G 2 . All constraints (1a) are satisfied. The blossom constraints (1b) act only on x-variables, i.e., on individual stages. Since our graph is bipartite, only considering the x-variables of one stage and disregarding (1b) yields the bipartite matching polytope which has only integral vertices; our (sub)solution is an element of this polytope. Thus, (1b) cannot be violated by our assignment.

Approximation
We start with the special case of 2-IM, before extending the result to the multistage MIM scenario. Then we will transform the algorithms for use with 2-UM and MUM.

Approximating 2-IM
We first describe Algorithm 1, which is an approximation for 2-IM. Although its ratio is not constant but grows with the rate of √ μ, Theorem 7 hints that better approximations may be hard to obtain. Algorithm 1 roughly works as follows: Given a 2-stage graph G, we iterate the following procedure on G 1 until every edge of E ∩ has been in at least one perfect matching: Compute a perfect matching M 1 in G 1 that uses the maximum number of edges of E ∩ that have not been used in any previous iteration; then compute a perfect matching M 2 in G 2 that optimizes the profit with respect to M 1 . While doing so, keep track of the maximal occurring profit. Note that by choosing weights appropriately, we can construct a perfect matching that contains the maximum number of edges of some prescribed edge set in polynomial time [19]. We show: We prove this via two Lemmata; the bad instance of Lemma 9 in conjunction with the approximation guarantee (Lemma 10) establishes tightness.

Lemma 9 (Bad instance) The approximation ratio of Algorithm
Proof Consider the following family G k of 2-IM instances, parameterized by number k ≥ 1. An example using k = 4 is depicted in Fig. 4 Observe that G k is reduced, its union graph is bipartite, and μ = k + i∈[k −1] By construction, for any perfect matching M in the first (second) stage, |M ∩ E ∩ | ≤ k. Let M F (M F ) denote the unique perfect matching that contains uu 0 (u u 0 , respectively) and all of F. The pair (M F , M F ) is an optimal solution with profit |F| = k.
Consider an alternative perfect matching M in the first stage. In each cycle C i , we consider the shared edge w i w i and its opposing edge (i.e., its unique non-adjacent edge in C i ). We distinguish between three possibilities regarding their memberships in M: the shared edge and its opposing edge are in M (type Y ), only the opposing edge is in M (type N1), none of them are in M (type N2).
Picking an edge adjacent to u determines a perfect matching up to the types of some C i -cycles. For i ∈ k − 1 , let M i denote the unique perfect matching that contains uu i , is type Y in C i+1 , but type N2 in each C j for j > i + 1. Note that M i is type N1 in each C j with j ≤ i and contains all i shared edges along P i . Thus, |M i ∩ E ∩ | = i + 1. Most importantly, consider any perfect matching M in the second stage. By construction, for any i ∈ k − 1 , no two edges of (M i ∩ A) ∪ {w i+1 w i+1 } can be contained simultaneously in M . It follows that |M i ∩ M | ≤ 1, Algorithm 1 may never choose the optimal M F as a perfect matching for the first stage: In the first iteration, both M F and M k−1 have weight k, so the algorithm may choose M k−1 and obtain a 2-stage perfect matching with profit 1. In the following iteration, the weight (denoting the preference of edges) of M F is decreased by 1, since the edge w k w k has already been chosen in M k−1 . Consequently, in each following iteration i ∈ [k] the algorithm may choose M k−i over M F , each time decreasing the weight of M F by 1. After choosing M 0 over M F in iteration k, each edge in E ∩ has 1 , R i and R * i for i ∈ [3] been in some matching in the first stage; the algorithm stops and returns a 2-stage perfect matching with profit 1.
Proof Let G be a feasible and reduced 2-stage graph with non-empty E ∩ . Clearly, our algorithm achieves apx ≥ 1 as described in In case opt ≤ q, any solution with profit at least 1 yields a (1/q)-approximation. We show that we are in one of these cases.
Let q := q . Assume that opt > q (thus opt ≥ q) and simultaneously r * i < opt/q for all i ∈ [k]. Since we distribute M * ∩ over the disjoint sets {R * i | i ∈ [k]}, each containing less than opt/q edges, we know that k > q (thus k ≥ q). In iteration i, M * 1 has weight |(M * Hence, the latter term is a lower bound on r i , that we estimate as follows: −1] opt/q = opt · 1 − (i − 1)/q . The above assumptions give a contradiction: and M τ to T τ // set initial solution 7 foreach i ∈ [τ − 1] do // modify solution according to M P 8 if e i ∈ M P then set M i to S i and M i+1 to T i+1

Approximating MIM
Let us extend the above result to an arbitrary number of stages. We show that we can use any 2-IM approximation algorithm (in particular also Algorithm 1) as a black box to obtain an approximation algorithm for MIM, while only halving the approximation ratio: Algorithm 2 uses an edge weighted path (P, w) on τ vertices as an auxiliary graph. We set the weight of the edge between the ith and (i + 1)th vertex to an approximate solution for the 2-IM instance that arises from the ith and (i + 1)th stage of the MIM instance. A maximum weight matching M P in (P, w) induces a feasible solution for the MIM problem: If an edge ( j, j +1) is in M P , we use the corresponding solutions for the jth and ( j + 1)th stage; for stages without incident edge in M P , we select an arbitrary solution. Since no vertex is incident to more than one edge in M P , there are no conflicts.
We compute a maximum weight matching in a path in linear time using straightforward dynamic programming. Hence, assuming running time f for A, Corollary 13 Algorithms 1 in Algorithm 2 yields a (1/ √ 8μ)-approximation for MIM.
There is another way to approximate MIM via an approximation for 2-IM, which neither dominates nor is dominated by the above method:

Theorem 14
There is an S-reduction from MIM to 2-IM, i.e., given any MIM instance G, we can find a corresponding 2-IM instance G in polynomial time such that any solution for G bijectively corresponds to a solution for G with the same profit. Furthermore, Proof We will construct a 2-stage graph G whose first stage G 1 consists of (subdivided) disjoint copies of G i for odd i; conversely its second stage G 2 consists of (subdivided) disjoint copies of G i for even i. More precisely, consider the following construction: Let b(i) := 2 − (i mod 2). For each i ∈ [τ ], we create a copy of G i in G b(i) where each edge e ∈ E(G i ) is replaced by a 7-path p e i . We label the 3rd (5th) edge along p e i (disregarding its orientation) with e − i (e + i , respectively). To finally obtain G , for each i ∈ [τ − 1] and e ∈ E(G i ) ∩ E(G i+1 ), we identify the vertices of e + i with those of e − i+1 (disregarding the edges' orientations); thereby precisely the edges e + i and e − i+1 become an edge common to both stages. No other edges are shared between both stages. This completes the construction of G and we have Assume the approximation ratio for 2-IM would not depend on μ. Then the above would yield a surprisingly strong result:

Approximating MUM
Consider the MUM-problem which minimizes the cost. As noted in Observation 4, a 2-approximation is easily accomplished. However, by exploiting the previous results for MIM, we obtain better approximations.
Note that a similar reduction from MIM to MUM is not achieved as easily: Consider any (1 + ε)-approximation for MUM. Choose an even integer k ≥ 6 such that k/(k − 1) ≤ 1 + ε; consider a spanning 2-stage instance where each stage is a k-cycle and E ∩ consists of a single edge e. The optimal 2-stage perfect matching M * that contains e in both stages has profit p(M * ) = 1 and cost c(M * ) = 2 · k/2 − 1 = k − 1. A 2-stage perfect matching M that does not contain e still satisfies c(M) = k and as such is an (1 + ε)-approximation for MUM. However, its profit p(M) = 0 does not provide any approximation of p(M * ) = 1.
As for MIM, we aim to extend a given approximation for 2-UM to a general approximation for MUM. Unfortunately, we cannot use Theorems 14 and 17 for this, as an approximation for 2-UM does not generally constitute one for 2-IM (and MIM). On the positive side, a similar approach as used in the proof of Theorem 12 also works for minimization. Since σ (I ∪ J ) α-approximates the sum of all 2-UM instances' solution values, we have σ (I ∪ J ) ≤ α · opt. For each transition, any solution satisfies (n i + n i+1 )/4 ≤ |M i ∪ M i+1 | and hence ξ(I ∪ J ) ≤ 2 · opt. Finally, we obtain the claimed ratio: apx ≤ 1/2 · 2 · opt + α · opt = (1 + α/2) · opt.

Conclusion
In this paper we presented the first approximation algorithm for 2-IM with a (tightly analyzed) approximation ratio of 1/ √ 2μ. It remains open if a constant factor approximation for 2-IM is possible; however, we showed that this would imply a constant factor approximation for MIM. We further showed two ways in which MIM and MUM can be approximated by using any algorithm that approximates 2-IM, thereby also presenting the first approximation algorithms for multistage matching problems with an arbitrary number of stages.
Furthermore, it turns out that the techniques of Algorithms 1 and 2 for MIM can be used as a building block to approximate a broad set of related problems, so-called Multistage Subgraph Problems [12].
Funding Open Access funding enabled and organized by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. We are not aware of any way to circumvent this problem.