maximum-weight-matching/doc/Algorithm.md

Implementation of Maximum Weighted Matching

Introduction

This document describes the implementation of an algorithm that computes a maximum weight matching in a general graph in time O(n³), where n is the number of vertices in the graph.

In graph theory, a matching is a subset of edges such that none of them share a common vertex.

A maximum cardinality matching is a matching that contains the largest possible number of edges (or equivalently, the largest possible number of vertices).

For a graph that has weights attached to its edges, a maximum weight matching is a matching that achieves the largest possible sum of weights of matched edges. An algorithm for maximum weight matching can obviously also be used to calculate a maximum cardinality matching by simply assigning weight 1 to all edges.

Certain computer science problems can be understood as restrictions of maximum weighted matching in general graphs. Examples are: maximum matching in bipartite graphs, maximum cardinality matching in general graphs, and maximum weighted matching in general graphs with edge weights limited to integers in a certain range. Clearly, an algorithm for maximum weighted matching in general graphs also solves all of these restricted problems. However, some of the restricted problems can be solved with algorithms that are simpler and/or faster than the known algorithms for the general problem. The rest of this document does not consider restricted problems. My focus is purely on maximum weighted matching in general graphs.

In this document, n refers to the number of vertices and m refers to the number of edges in the graph.

A timeline of matching algorithms

In 1963, Jack Edmonds published the first polynomial-time algorithm for maximum matching in general graphs [1] [2] . Efficient algorithms for bipartite graphs were already known at the time, but generalizations to non-bipartite graphs would tend to require an exponential number of steps. Edmonds solved this by explicitly detecting blossoms (odd-length alternating cycles) and adding special treatment for them. He also introduced a linear programming technique to handle weighted matching. The resulting maximum weighted matching algorithm runs in time O(n⁴).

In 1973, Harold N. Gabow published a maximum weighted matching algorithm that runs in time O(n³) [3] . It is based on the ideas of Edmonds, but uses different data structures to reduce the amount of work.

In 1983, Galil, Micali and Gabow published a maximum weighted matching algorithm that runs in time O(n*m*log(n)) [4] . It is an implementation of Edmonds' blossom algorithm that uses advanced data structures to speed up critical parts of the algorithm. This algorithm is asymptotically faster than O(n³) for sparse graphs, but slower for highly dense graphs.

In 1983, Zvi Galil published an overview of algorithms for 4 variants of the matching problem [5] : maximum-cardinality resp. maximum-weight matching in bipartite resp. general graphs. I like this paper a lot. It explains the algorithms from first principles and can be understood without prior knowledge of the literature. The paper describes a maximum weighted matching algorithm that is similar to Edmonds' blossom algorithm, but carefully implemented to run in time O(n³). It then sketches how advanced data structures can be added to arrive at the Galil-Micali-Gabow algorithm that runs in time O(n*m*log(n)).

In 1990, Gabow published a maximum weighted matching algorithm that runs in time O(n*m + n²*log(n)) [6] . It uses several advanced data structures, including Fibonacci heaps. Unfortunately I don't understand this algorithm at all.

Choosing an algorithm

I selected the O(n³) variant of Edmonds' blossom algorithm as described by Galil [5] . This algorithm is usually credited to Gabow [3], but I find the description in [5] easier to understand.

This is generally a fine algorithm. One of its strengths is that it is relatively easy to understand, especially compared to the more recent algorithms. Its run time is asymptotically optimal for complete graphs (graphs that have an edge between every pair of vertices).

On the other hand, the algorithm is suboptimal for sparse graphs. It is possible to construct highly sparse graphs, having m = O(n), that cause this algorithm to perform Θ(n³) steps. In such cases the Galil-Micali-Gabow algorithm would be significantly faster.

Then again, there is a question of how important the worst case is for practical applications. I suspect that the simple algorithm typically needs only about O(n*m) steps when running on random sparse graphs with random weights, i.e. much faster than its worst case bound.

After trading off these properties, I decided that I prefer an algorithm that is understandable and has decent performance, over an algorithm that is faster in specific cases but also significantly more complicated.

Description of the algorithm

My implementation closely follows the description by Zvi Galil in [5] . I recommend to read that paper before diving into my description below. The paper explains the algorithm in depth and shows how it relates to matching in bipartite graphs and non-weighted graphs.

There are subtle aspects to the algorithm that are tricky to implement correctly but are mentioned only briefly in the paper. In this section, I describe the algorithm from my own perspective: a programmer struggling to implement the algorithm correctly.

My goal is only to describe the algorithm, not to prove its correctness.

Basic concepts

An edge-weighted, undirected graph G consists of a set V of n vertices and a set E of m edges.

Vertices are represented by non-negative integers from 0 to n-1: V = { 0, 1, ..., n-1 }

Edges are represented by tuples: E = { (x, y, w), ... }
where the edge (x, y, w) is incident on vertices x and y and has weight w.
The order of vertices is irrelevant, i.e. (x, y, w) and (y, x, w) represent the same edge.
Edge weights may be integers or floating point numbers.
There can be at most one edge between any pair of vertices.
A vertex can not have an edge to itself.

A matching is a subset of edges without any pair of edges sharing a common vertex.
An edge is matched if it is part of the matching, otherwise it is unmatched.
A vertex is matched if it is incident to an edge in the matching, otherwise it is unmatched.

An alternating path is a simple path that alternates between matched and unmatched edges.

Figure 1

Figure 1 depicts a graph with 6 vertices and 9 edges. Wavy lines represent matched edges; straight lines represent edges that are currently unmatched. The matching has weight 8 + 7 = 15. An example of an alternating path would be 0 - 1 - 4 - 5 - 3 (but there are many others).

Augmenting paths

An augmenting path is an alternating path that begins and ends in two unmatched vertices.

An augmenting path can be used to extend the matching as follows: remove all matched edges on the augmenting path from the matching, and add all previously unmatched edges on the augmenting path to the matching. The result is again a valid matching, and the number of matched edges has increased by one.

The matching in figure 1 above has several augmenting paths. For example, the edge from vertex 0 to vertex 2 is by itself an augmenting path. Augmenting along this path would increase the weight of the matching by 2. Another augmenting path is 0 - 1 - 4 - 5 - 3 - 2, which would increase the weight of the matching by 3. Finally, 0 - 4 - 1 - 2 is also an augmenting path, but it would decrease the weight of the matching by 2.

Main algorithm

Our algorithm to compute a maximum weighted matching is as follows:

• Start with an empty matching (all edges unmatched).
• Repeat the following steps:
  • Find an augmenting path that provides the largest possible increase of the weight of the matching.
  • If there is no augmenting path that increases the weight of the matching, end the algorithm. The current matching is a maximum-weight matching.
  • Otherwise, use the augmenting path to update the matching.
  • Continue by searching for another augmenting path, etc.

This algorithm ends when there are no more augmenting paths that increase the weight of the matching. In some cases, there may still be augmenting paths which do not increase the weight of the matching, implying that the maximum-weight matching has fewer edges than the maximum cardinality matching.

Every iteration of the main loop is called a stage. Note that the final matching contains at most n/2 edges, therefore the algorithm performs at most n/2 stages.

The only remaining challenge is finding an augmenting path. Specifically, finding an augmenting path that increases the weight of the matching as much as possible.

Blossoms

A blossom is an odd-length cycle that alternates between matched and unmatched edges. Such cycles complicate the search for augmenting paths. To overcome these problems, blossoms must be treated specially. The trick is to temporarily replace the vertices and edges that are part of the blossom by a single super-vertex. This is called shrinking the blossom. The search for an augmenting path then continues in the modified graph in which the odd-length cycle no longer exists. It may later become necessary to expand the blossom (undo the shrinking step).

For example, the cycle 0 - 1 - 4 - 0 in figure 1 above is an odd-length alternating cycle, and therefore a candidate to become a blossom.

In practice, we do not remove vertices or edges from the graph while shrinking a blossom. Instead the graph is left unchanged, but a separate data structure keeps track of blossoms and which vertices are contained in which blossoms.

A graph can contain many blossoms. Furthermore, after shrinking a blossom, that blossom can become a sub-blossom in a bigger blossom. Figure 2 depicts a graph with several nested blossoms.

Figure 2: Nested blossoms

To describe the algorithm unambiguously, we need precise definitions:

• A blossom is either a single vertex, or an odd-length alternating cycle of sub-blossoms.
• A non-trivial blossom is a blossom that is not a single vertex.
• A top-level blossom is a blossom that is not contained inside another blossom. It can also be a single vertex which is not part of any blossom.
• The base vertex of a blossom is the only vertex in the blossom that is not matched to another vertex in the same blossom. The base vertex is either unmatched, or matched to a vertex outside the blossom. In a single-vertex blossom, its only vertex is the base vertex. In a non-trivial blossom, the base vertex is equal to the base vertex of the unique sub-blossom that begins and ends the alternating cycle.
• The parent of a non-top-level blossom is the directly enclosing blossom in which it occurs as a sub-blossom. Top-level blossoms do not have parents.

Non-trivial blossoms are created by shrinking and destroyed by expanding. These are explicit steps of the algorithm. This implies that not every odd-length alternating cycle is a blossom. The algorithm decides which cycles are blossoms.

Just before a new blossom is created, its sub-blossoms are initially top-level blossoms. After creating the blossom, the sub-blossoms are no longer top-level blossoms because they are contained inside the new blossom, which is now a top-level blossom.

Every vertex is contained in precisely one top-level blossom. This may be either a trivial blossom which contains only that single vertex, or a non-trivial blossom into which the vertex has been absorbed through shrinking. A vertex may belong to different top-level blossoms over time as blossoms are created and destroyed by the algorithm. I use the notation B(x) to indicate the top-level blossom that contains vertex x.

Searching for an augmenting path

Recall that the matching algorithm repeatedly searches for an augmenting path that increases the weight of the matching as much as possible. I'm going to postpone the part that deals with "increasing the weight as much as possible". For now, all we need to know is this: At a given step in the algorithm, some edges in the graph are tight while other edges have slack. An augmenting path that consists only of tight edges is guaranteed to increase the weight of the matching as much as possible.

While searching for an augmenting path, we simply restrict the search to tight edges, ignoring all edges that have slack. Certain explicit actions of the algorithm cause edges to become tight or slack. How this works will be explained later.

To find an augmenting path, the algorithm searches for alternating paths that start in an unmatched vertex. The collection of alternating paths forms a forest of trees. Each tree is rooted in an unmatched vertex, and all paths from the root to the leaves of a tree are alternating paths. The nodes in these trees are top-level blossoms.

To facilitate the search, top-level blossoms are labeled as either S or T or unlabeled. Label S is assigned to the roots of the alternating trees, and to nodes at even distance from the root. Label T is assigned to nodes at odd distance from the root of an alternating tree. Unlabeled blossoms are not (yet) part of an alternating tree. (In some papers, the label S is called outer and T is called inner.)

It is important to understand that labels S and T are assigned only to top-level blossoms, not to individual vertices. However, it is often relevant to know the label of the top-level blossom that contains a given vertex. I use the following terms for convenience:

• an S-blossom is a top-level blossom with label S;
• a T-blossom is a top-level blossom with label T;
• an S-vertex is a vertex inside an S-blossom;
• a T-vertex is a vertex inside a T-blossom.

Edges that span between two S-blossoms are special. If both S-blossoms are part of the same alternating tree, the edge is part of an odd-length alternating cycle. The lowest common ancestor node in the alternating tree forms the beginning and end of the alternating cycle. In this case a new blossom must be created by shrinking the cycle. If the two S-blossoms are in different alternating trees, the edge that links the blossoms is part of an augmenting path between the roots of the two trees.

Figure 3: Growing alternating trees

The graph in figure 3 contains two unmatched vertices: 0 and 7. Both form the root of an alternating tree. Blue arrows indicate edges in the alternating trees, pointing away from the root. The graph edge between vertices 4 and 6 connects two S-blossoms in the same alternating tree. Scanning this edge will cause the creation of a new blossom with base vertex 2. The graph edge between vertices 6 and 9 connects two S-blossoms which are part of different alternating trees. Scanning this edge will discover an augmenting path between vertices 0 and 7.

Note that all vertices in the graph of figure 3 happen to be top-level blossoms. In general, the graph may already contain non-trivial blossoms. Alternating trees are constructed over top-level blossoms, not necessarily over individual vertices.

When a vertex becomes an S-vertex, it is added to a queue. The search procedure considers these vertices one-by-one and tries to use them to either grow the alternating tree (thus adding new vertices to the queue), or discover an augmenting path or a new blossom.

In detail, the search for an augmenting path proceeds as follows:

• Mark all top-level blossoms as unlabeled.
• Initialize an empty queue Q.
• Assign label S to all top-level blossoms that contain an unmatched vertex. Add all vertices inside such blossoms to Q.
• Repeat until Q is empty:
  • Take a vertex x from Q.
  • Scan all tight edges (x, y, w) that are incident on vertex x.
    • Find the top-level blossom B(x) that contains vertex x and the top-level blossom B(y) that contains vertex y.
    • If B(x) and B(y) are the same blossom, ignore this edge. (It is an internal edge in the blossom.)
    • Otherwise, if blossom B(y) is unlabeled, assign label T to it. The base vertex of B(y) is matched (otherwise B(y) would have label S). Find the vertex t which is matched to the base vertex of B(y) and find its top-level blossom B(t) (this blossom is still unlabeled). Assign label S to blossom B(t), and add all vertices inside B(t) to Q.
    • Otherwise, if blossom B(y) has label T, ignore this edge.
    • Otherwise, if blossom B(y) has label S, there are two scenarios:
      • Either B(x) and B(y) are part of the same alternating tree. In that case, we have discovered an odd-length alternating cycle. Shrink the cycle into a blossom and assign label S to it. Add all vertices inside its T-labeled sub-blossoms to Q. (Vertices inside S-labeled sub-blossoms have already been added to Q and must not be added again.) Continue the search for an augmenting path.
      • Otherwise, if B(x) and B(y) are part of different alternating trees, we have found an augmenting path between the roots of those trees. End the search and return the augmenting path.
• If Q becomes empty before an augmenting path has been found, it means that no augmenting path exists (which consists of only tight edges).

For each top-level blossom, we keep track of its label as well as the edge through which it obtained its label (attaching it to the alternating tree). These edges are needed to trace back through the alternating tree to construct a blossom or an alternating path.

When an edge between S-blossoms is discovered, it is handled as follows:

• The two top-level blossoms are B(x) and B(y), both labeled S.
• Trace up through the alternating trees from B(x) and B(y) towards the root.
• If the blossoms are part of the same alternating tree, the tracing process eventually reaches a lowest common ancestor of B(x) and B(y). In that case a new blossom must be created. Its alternating cycle starts at the common ancestor, follows the path through the alternating tree down to B(x), then via the scanned edge to B(y), then through the alternating tree up to the common ancestor. (Note it is possible that either B(x) or B(y) is itself the lowest common ancestor.)
• Otherwise, the blossoms are in different trees. The tracing process will eventually reach the roots of both trees. At that point we have traced out an augmenting path between the two roots.

Note that the matching algorithm uses two different types of tree data structures. These two types of trees are separate concepts. But since they are both trees, there is potential for accidentally confusing them. It may be helpful to explicitly distinguish the two types:

• A forest of blossom structure trees represents the nested structure of blossoms. Every node in these trees is a blossom. The roots are the top-level blossoms. The leaf nodes are the single vertices. The child nodes of each non-leaf node represent its sub-blossoms. Every vertex is a leaf node in precisely one blossom structure tree. A blossom structure tree may consist of just one single node, representing a vertex that is not part of any non-trivial blossom.
• A forest of alternating trees represents an intermediate state during the search for an augmenting path. Every node in these trees is a top-level blossom. The roots are top-level blossoms with an unmatched base vertex. Every labeled top-level blossom is part of an alternating tree. Unlabeled blossoms are not (yet) part of a tree.

Augmenting the matching

Once an augmenting path has been found, augmenting the matching is relatively easy. Simply follow the path, adding previously unmatched edges to the matching and removing previously matched edges from the matching.

A useful data structure to keep track of matched edges is an array mate[x] indexed by vertex. If vertex x is matched, mate[x] contains the index of the vertex to which it is matched. If vertex x is unmatched, mate[x] contains -1.

A slight complication arises when the augmenting path contains a non-trivial blossom. The search returns an augmenting path over top-level blossoms, without details about the layout of the path within blossoms. Any parts of the path that run through a blossom must be traced in order to update the matched/unmatched status of the edges in the blossom.

When an augmenting path runs through a blossom, it always runs between the base vertex of the blossom and some sub-blossom (potentially the same sub-blossom that contains the base vertex). If the base vertex is unmatched, it forms the start or end of the augmenting path. Otherwise, the augmenting path enters the blossom though the matched edge of the base vertex. From the opposite direction, the augmenting path enters the blossom through an unmatched edge. It follows that the augmenting path must run through an even number of internal edges of the blossom. Fortunately, every sub-blossom can be reached from the base through an even number of steps by walking around the blossom in the appropriate direction.

Augmenting along a path through a blossom causes a reorientation of the blossom. Afterwards, it is still a blossom and still consists of an odd-length alternating cycle, but the cycle begins and ends in a different sub-blossom. The blossom also has a different base vertex. (In specific cases where the augmenting path merely "grazes" a blossom, the orientation and base vertex remain unchanged.)

Figure 4: Augmenting path through a blossom

Figure 4 shows an augmenting path that runs through a blossom. The augmenting path runs through an even number of internal edges in the blossom, which in this case is not the shortest way around the blossom. After augmenting, the blossom has become reoriented: a different vertex became the base vertex.

In case of nested blossoms, non-trivial sub-blossoms on the augmenting path must be updated recursively.

Note that the process of repeatedly augmenting the matching will never cause a matched vertex to become unmatched. Once a vertex is matched, augmenting may cause the vertex to become matched to a different vertex, but it can not cause the vertex to become unmatched.

Edge slack and dual variables

We still need a method to determine which edges are tight. This is done by means of so-called dual variables.

The purpose of dual variables can be explained by rephrasing the maximum matching problem as an instance of linear programming. I'm not going to do that here. I will describe how the algorithm uses dual variables without explaining why. For the mathematical background, I recommend reading [5] .

Every vertex x has a dual variable u_x. Furthermore, every non-trivial blossom B has a dual variable z_B. These variables contain non-negative numbers which change over time through actions of the algorithm.

Every edge in the graph imposes a constraint on the dual variables: The weight of the edge between vertices x and y must be less or equal to the sum of the dual variables u_x plus u_y plus all z_B of blossoms B that contain the edge. (A blossom contains an edge if it contains both incident vertices.) This constraint is more clearly expressed in a formula:

 u_x + u_y + \sum_{(x,y) \in B} z_B \ge w_{x,y} 

The slack π_x,y of the edge between vertices x and y is a non-negative number that indicates how much room is left before the edge constraint would be violated:

 \pi_{x,y} = u_x + u_y + \sum_{(x,y) \in B} z_B - w_{x,y} 

An edge is tight if and only if its slack is zero. Given the values of the dual variables, it is very easy to calculate the slack of an edge which is not contained in any blossom: simply add the duals of its incident vertices and subtract the weight. To check whether an edge is tight, simply compute its slack and check whether it is zero.

Calculating the slack of an edge that is contained in one or more blossoms is a little tricky, but fortunately we don't need such calculations. The search for augmenting paths only considers edges that span between top-level blossoms, not edges that are contained inside blossoms. So we never need to check the tightness of internal edges in blossoms.

A matching has maximum weight if it satisfies all of the following constraints:

• All dual variables and edge slacks are non-negative: u_i ≥ 0 , z_B ≥ 0 , π_x,y ≥ 0
• All matched edges have zero slack: edge (x, y) matched implies π_x,y = 0
• All unmatched vertices have dual variable zero: vertex x unmatched implies u_x = 0

The first two constraints are satisfied at all times while the matching algorithm runs. When the algorithm updates dual variables, it ensures that dual variables and edge slacks remain non-negative. It also ensures that matched edges remain tight, and that edges which are part of the odd-length cycle in a blossom remain tight. When the algorithm augments the matching, it uses an augmenting path that consists of only tight edges, thus ensuring that newly matched edges have zero slack.

The third constraint is initially not satisfied. The algorithm makes progress towards satisfying this constraint in two ways: by augmenting the matching, thus reducing the number of unmatched vertices, and by reducing the value of the dual variables of unmatched vertices. Eventually, either all vertices are matched or all unmatched vertices have zero dual. At that point the maximum weight matching has been found.

When the matching algorithm is finished, the constraints can be checked to verify that the matching is optimal. This check is simpler and faster than the matching algorithm itself. It can therefore be a useful way to guard against bugs in the matching algorithm.

Rules for updating dual variables

At the start of the matching algorithm, all vertex dual variables u_i are initialized to the same value: half of the greatest edge weight value that occurs on any edge in the graph.

 u_i = {1 \over 2} \cdot \max_{(x,y) \in E} w_{x,y} 

Initially, there are no blossoms yet so there are no z_B to be initialized. When the algorithm creates a new blossom, it initializes its dual variable to z_B = 0. Note that this does not change the slack of any edge.

If a search for an augmenting path fails while there are still unmatched vertices with positive dual variables, it may not yet have found the maximum weight matching. In such cases the algorithm updates the dual variables until either an augmenting path gets unlocked or the dual variables of all unmatched vertices reach zero.

To update the dual variables, the algorithm chooses a value δ that represents how much the duals will change. It then changes dual variables as follows:

• u_x ← u_x − δ for every S-vertex x
• u_x ← u_x + δ for every T-vertex x
• z_B ← z_B + 2 * δ for every non-trivial S-blossom B
• z_B ← z_B − 2 * δ for every non-trivial T-blossom B

Dual variables of unlabeled blossoms and their vertices remain unchanged. Dual variables z_B of non-trivial sub-blossoms also remain changed; only top-level blossoms have their z_B updated.

Note that this update does not change the slack of edges that are either matched, or linked in the alternating tree, or contained in a blossom. However, the update reduces the slack of edges between S blossoms and edges between S-blossoms and unlabeled blossoms. It may cause some of these edges to become tight, allowing them to be used to construct an augmenting path.

The value of δ is determined as follows: δ = min(δ₁, δ₂, δ₃, δ₄) where

• δ₁ is the minimum u_x of any S-vertex x.
• δ₂ is the minimum slack of any edge between an S-blossom and an unlabeled blossom.
• δ₃ is half of the minimum slack of any edge between two different S-blossoms.
• δ₄ is half of the minimum z_B of any T-blossom B.

δ₁ protects against any vertex dual becoming negative. δ₂ and δ₃ together protect against any edge slack becoming negative. δ₄ protects against any blossom dual becoming negative.

If the dual update is limited by δ₁, it causes the duals of all remaining unmatched vertices to become zero. At that point the maximum matching has been found and the algorithm ends. If the dual update is limited by δ₂ or δ₃, it causes an edge to become tight. The next step is to either add that edge to the alternating tree (δ₂) or use it to construct a blossom or augmenting path (δ₃). If the dual update is limited by δ₄, it causes the dual variable of a T-blossom to become zero. The next step is to expand that blossom.

A dual update may find that δ = 0, implying that the dual variables don't change. This can still be useful since all types of updates have side effects (adding an edge to an alternating tree, or expanding a blossom) that allow the algorithm to make progress.

During a single stage, the algorithm may iterate several times between scanning tight edges and updating dual variables. These iterations are called substages. To clarify: A stage is the process of growing alternating trees until an augmenting path is found. A stage ends either by augmenting the matching, or by concluding that no further improvement is possible. Each stage consists of one or more substages. A substage scans tight edges to grow the alternating trees. When a substage runs out of tight edges, it ends by performing a dual variable update. A substage also ends immediately when it finds an augmenting path. At the end of each stage, labels and alternating trees are removed.

The matching algorithm ends when a substage ends in a dual variable update limited by δ₁. At that point the matching has maximum weight.

Expanding a blossom

There are two scenarios where a blossom must be expanded. One is when the dual variable of a T-blossom becomes zero after a dual update limited by δ₄. In this case the blossom must be expanded, otherwise further dual updates would cause its dual to become negative.

The other scenario is when the algorithm is about to assign label T to an unlabeled blossom with dual variable zero. A T-blossom with zero dual serves no purpose, potentially blocks an augmenting path, and is likely to be expanded anyway via a δ₄=0 update. It is therefore better to preemptively expand the unlabeled blossom. The step that would have assigned label T to the blossom is then re-evaluated, which will cause it to assign label T to a sub-blossom of the expanded blossom. It may then turn out that this sub-blossom must also be expanded, and this becomes a recursive process until we get to a sub-blossom that is either a single vertex or a blossom with non-zero dual.

Note that [5] specifies that all top-level blossoms with dual variable zero should be expanded after augmenting the matching. This prevents the existence of unlabeled top-level blossoms with zero dual, and therefore prevents the scenario where label T would be assigned to such blossoms. That strategy is definitely correct, but it may lead to unnecessary expanding of blossoms which are then recreated during the search for the next augmenting path. Postponing the expansion of these blossoms until they are about to be labeled T, as described above, may be faster in some cases.

Expanding an unlabeled top-level blossom B is pretty straightforward. Simply promote all of its sub-blossoms to top-level blossoms, then delete B. Note that the slack of all edges remains unchanged, since z_B = 0.

Expanding a T-blossom is tricky because the labeled blossom is part of an alternating tree. After expanding the blossom, the part of the alternating tree that runs through the blossom must be reconstructed. An alternating path through a blossom always runs through its base vertex. After expanding T-blossom B, we can reconstruct the alternating path by following it from the sub-blossom where the path enters B to the sub-blossom that contains the base vertex (choosing the direction around the blossom that takes an even number of steps). We then assign alternating labels T and S to the sub-blossoms along that path and link them into the alternating tree. All vertices of sub-blossoms that got label S are inserted into Q.

Figure 5: Expanding a T-blossom

Keeping track of least-slack edges

To perform a dual variable update, the algorithm needs to compute the values of δ₁, δ₂, δ₃ and δ₄ and determine which edge (δ₂, δ₃) or blossom (δ₄) achieves the minimum value.

The total number of dual updates during a matching may be Θ(n²). Since we want to limit the total number of steps of the matching algorithm to O(n³), each dual update may use at most O(n) steps.

We can find δ₁ in O(n) steps by simply looping over all vertices and checking their dual variables. We can find δ₄ in O(n) steps by simply looping over all non-trivial blossoms (since there are fewer than n non-trivial blossoms). We could find δ₂ and δ₃ by simply looping over all edges of the graph in O(m) steps, but that exceeds our budget of O(n) steps. So we need better techniques.

For δ₂, we determine the least-slack edge between an S-blossom and unlabeled blossom as follows. For every vertex y in any unlabeled blossom, keep track of e_y: the least-slack edge that connects y to any S-vertex. The thing to keep track of is the identity of the edge, not the slack value. This information is kept up-to-date as part of the procedure that considers S-vertices. The scanning procedure eventually considers all edges (x, y, w) where x is an S-vertex. At that moment e_y is updated if the new edge has lower slack.

Calculating δ₂ then becomes a matter of looping over all vertices x, checking whether B(x) is unlabeled and calculating the slack of e_x.

One subtle aspect of this technique is that a T-vertex can loose its label when the containing T-blossom gets expanded. At that point, we suddenly need to have kept track of its least-slack edge to any S-vertex. It is therefore necessary to do the same tracking also for T-vertices. So the technique is: For any vertex that is not an S-vertex, track its least-slack edge to any S-vertex.

Another subtle aspect is that a T-vertex may have a zero-slack edge to an S-vertex. Even though these edges are already tight, they must still be tracked. If the T-vertex loses its label, this edge needs to be reconsidered by the scanning procedure. By including these edges in least-slack edge tracking, they will be rediscovered through a δ₂=0 update after the vertex becomes unlabeled.

For δ₃, we determine the least-slack edge between any pair of S-blossoms as follows. For every S-blossom B, keep track of e_B: the least-slack edge between B and any other S-blossom. Note that this is done per S-blossoms, not per S-vertex. This information is kept up-to-date as part of the procedure that considers S-vertices. Calculating δ₃ then becomes a matter of looping over all S-blossoms B and calculating the slack of e_B.

A complication occurs when S-blossoms are merged. Some of the least-slack edges of the sub-blossoms may be internal edges in the merged blossom, and therefore irrelevant for δ₃. As a result, the proper e_B of the merged blossom may be different from all least-slack edges of its sub-blossoms. An additional data structure is needed to find e_B of the merged blossom.

For every S-blossom B, maintain a list L_B of edges between B and other S-blossoms. The purpose of L_B is to contain, for every other S-blossom, the least-slack edge between B and that blossom. These lists are kept up-to-date as part of the procedure that considers S-vertices. While considering vertex x, if edge (x, y, w) has positive slack, and B(y) is an S-blossom, the edge is inserted in L_B(x). This may cause L_B(x) to contain multiple edges to B(y). That's okay as long as it definitely contains the least-slack edge to B(y).

When a new S-blossom is created, form its list L_B by merging the lists of its sub-blossoms. Ignore edges that are internal to the merged blossom. If there are multiple edges to the same target blossom, keep only the least-slack of these edges. Then find e_B of the merged blossom by simply taking the least-slack edge out of L_B.

Run time of the algorithm

Every stage of the algorithm either increases the number of matched vertices by 2 or ends the matching. Therefore the number of stages is at most n/2. Every stage runs in O(n²) steps, therefore the complete algorithm runs in O(n³) steps.

During each stage, edge scanning is driven by the queue Q. Every vertex enters Q at most once. Each vertex that enters Q has its incident edges scanned, causing every edge in the graph to be scanned at most twice per stage. Scanning an edge is done in constant time, unless it leads to the discovery of a blossom or an augmenting path, which will be separately accounted for. Therefore edge scanning needs O(m) steps per stage.

Creating a blossom reduces the number of top-level blossoms by at least 2, thus limiting the number of simultaneously existing blossoms to O(n). Blossoms that are created during a stage become S-blossoms and survive until the end of that stage, therefore O(n) blossoms are created during a stage. Creating a blossom involves tracing the alternating path to the closest common ancestor, and some bookkeeping per sub-blossom, and inserting new S-vertices Q, all of which can be done in O(n) steps per blossom creation. The cost of managing least-slack edges between S-blossoms will be separately accounted for. Therefore blossom creation needs O(n²) steps per stage (excluding least-slack edge management).

As part of creating a blossom, a list L_B of least-slack edges must be formed. This involves processing every element of all least-slack edge lists of its sub-blossoms, and removing redundant edges from the merged list. Merging and removing redundant edges can be done in one sweep via a temporary array indexed by target blossom. Collect the least-slack edges of the sub-blossoms into this array, indexed by the target blossom of the edge, keeping only the edge with lowest slack per target blossom. Then convert the array back into a list by removing unused indices. This takes O(1) steps per candidate edge, plus O(n) steps to manage the temporary array. I choose to shift the cost of collecting the candidate edges from the sub-blossoms to the actions that inserted those edges into the sub-blossom lists. There are two processes which insert edges into L_B: edge scanning and blossom creation. Edge scanning inserts each graph edge at most twice per stage for a total cost of O(m) steps per stage. Blossom creation inserts at most O(n) edges per blossom creation. Therefore the total cost of S-blossom least-slack edge management is O(m + n²) = O(n²) steps per stage.

The number of blossom expansions during a stage is O(n). Expanding a blossom involves some bookkeeping per sub-blossom, and reconstructing the alternating path through the blossom, and inserting any new S-vertices into Q, all of which can be done in O(n) steps per blossom. Therefore blossom expansion needs O(n²) steps per stage.

The length of an augmenting path is O(n). Tracing the augmenting path and augmenting the matching along the path can be done in O(n) steps. Augmenting through a blossom takes a number of steps that is proportional in the number of its sub-blossoms. Since there are fewer than n non-trivial blossoms, the total cost of augmenting through blossoms is O(n) steps. Therefore the total cost of augmenting is O(n) steps per stage.

A dual variable update limited by δ₁ ends the algorithm and therefore happens at most once. An update limited by δ₂ labels a previously labeled blossom and therefore happens O(n) times per stage. An update limited by δ₃ either creates a blossom or finds an augmenting path and therefore happens O(n) times per stage. An update limited by δ₄ expands a blossom and therefore happens O(n) times per stage. Therefore the number of dual variable updates is O(n) per stage. The cost of calculating the δ values is O(n) per update as discussed above. Applying changes to the dual variables can be done by looping over all vertices and looping over all top-level blossoms in O(n) steps per update. Therefore the total cost of dual variable updates is O(n²) per stage.

Implementation details

This section describes some choices I made while implementing the algorithm. There may be easier or faster or better ways to do it, but this is what I did and it seems to work mostly okay.

Data structures

Input graph

Vertices are represented as non-negative integers in range 0 to n-1.

Edges are represented as an array of tuples (x, y, w) where x and y are indices of the incident vertices, and w is the numerical weight of the edge. The edges are listed in no particular order. Each edge has an index in e range 0 to m-1.

edges[e] = (x, y, w)

In addition, edges are organized in an array of adjacency lists, indexed by vertex index. Each adjacency list contains the edge indices of all edges incident on a specific vertex. Every edge therefore appears in two adjacency lists.

adjacent_edges[x] = [e1, e2, ...]

These data structures are initialized at the start of the matching algorithm and never change while the algorithm runs.

General data

vertex_mate[x] = y if the edge between vertex x and vertex y is matched.
vertex_mate[x] = -1 if vertex x is unmatched.

vertex_top_blossom[x] = pointer to B(x), the top-level blossom that contains vertex x.

vertex_dual[x] holds the value of u_x.

A FIFO queue holds vertex indices of S-vertices whose edges have not yet been scanned. Vertices are inserted in this queue as soon as their top-level blossom gets label S.

Blossoms

A blossom is either a single vertex or a non-trivial blossom. Both types of blossoms are represented as class instances with the following attributes:

• B.base_vertex is the vertex index of the base vertex of blossom B.
• B.parent is a pointer to the parent of B in the blossom structure tree, or None if B is a top-level blossom.
• B.label is S or T or None
• B.tree_edge = (x, y) if B is a labeled top-level blossom, where y is a vertex in B and (x, y) is the edge that links B to its parent in the alternating tree.

A non-trivial blossom additionally has the following attributes:

• B.subblossoms is an array of pointers to the sub-blossoms of B, starting with the sub-blossom that contains the base vertex.
• B.edges is an array of alternating edges connecting the sub-blossoms.
• B.dual_var holds the value of z_B.

Single-vertex blossoms are kept in an array indexed by vertex index.
Non-trivial blossoms are kept in a separate array.
These arrays are used to iterate over blossoms and to find the trivial blossom that consists of a given vertex.

Least-slack edge tracking

vertex_best_edge[x] is an array holding e_x, the edge index of the least-slack edge between vertex x and any S-vertex, or -1 if there is no such edge. This value is only meaningful if x is a T-vertex or unlabeled vertex.

B.best_edge is a blossom attribute holding e_B, the edge index of the least-slack edge between blossom B and any other S-blossom, or -1 if there is no such edge. This value is only meaningful if B is a top-level S-blossom.

For non-trivial S-blossoms B, attribute B.best_edge_set holds the list L_B of potential least-slack edges to other blossoms. This list is not maintained for single-vertex blossoms, since L_B of a single vertex can be efficiently constructed from its adjacency list.

Memory usage

The data structures described above use a constant amount of memory per vertex and per edge and per blossom. Therefore the total memory requirement is O(m + n).

The memory usage of L_B is a little tricky. Any given list L_B can have length O(n), and O(n) of these lists can exist simultaneously. Naively allocating space for O(n) elements per list will drive memory usage up to O(n²). However, at any moment, an edge can be in at most two of these lists, therefore the sum of the lengths of these lists is limited to O(m). A possible solution is to implement the L_B as linked lists.

Performance critical routines

Calculations that happen very frequently in the algorithm are: determining the top-level blossom of a given vertex, and calculating the slack of a given edge. These calculations must run in constant time per call in any case, but it makes sense to put some extra effort into making these calculations fast.

Recursion

Certain tasks in the algorithm are recursive in nature: enumerating the vertices in a given blossom, and augmenting the matching along a path through a blossom. It seems natural to implement such tasks as recursive subroutines, which handle the task for a given blossom and make recursive calls to handle sub-blossoms as necessary. But the recursion depth of such implementations can grow to O(n) in case of deeply nested blossoms.

Deep recursion may cause problems in certain programming languages and runtime environments. In such cases, it may be better to avoid recursive calls and instead implement an iterative control flow with an explicit stack.

Handling integer edge weights

If all edge weights in the input graph are integers, it is possible and often desirable to implement the algorithm such that only integer calculations are used.

If all edge weights are integers, then all vertex dual variables u_x are integer multiples of 0.5, and all blossom dual variables z_B are integers. Storing the vertex duals as 2*u_x allows all calculations to be done with integers.

Proof by induction that all vertex duals are multiples of 0.5 and all blossom duals are integers:

• Vertex duals are initialized to 0.5 times the greatest edge weight. Blossom duals are initialized to 0. Therefore the proposition is initially true.
• All unmatched vertices have the same dual value. Proof: Initially all vertices have the same dual value. All unmatched vertices have been unmatched since the beginning, therefore always had label S in every dual variable update, therefore always got changed by the same amount.
• Either all duals of S-vertices are integers or all duals of S-vertices are odd multiples of 0.5. Proof: The root nodes of alternating trees are unmatched vertices which all have the same dual. Within an alternating tree, all edges are tight, therefore all duals of vertices in the tree differ by an integer amount, therefore either all duals are integers or all duals are odd multiples of 0.5.
• δ is a multiple of 0.5. Proof:
  • δ₁ = u_x and u_x is a multiple of 0.5, therefore δ₁ is a mutiple of 0.5.
  • δ₂ = π_x,y = u_x + u_y - w_x,y where u_x and u_y and w_x,y are multiples of 0.5, therefore δ₂ is a multiple of 0.5.
  • δ₃ = 0.5 * π_x,y = 0.5 * (u_x + u_y - w_x,y). Since x and y are S-vertices, either u_x and u_y are both integers or both are odd multiples of 0.5. In either case u_x + u_y is an integer. Therefore δ₃ is a multiple of 0.5.
  • δ₄ = 0.5 * z_B where z_B is an integer, therefore δ₄ is a multiple of 0.5.
• Vertex duals increase or decrease by δ which is a multiple of 0.5, therefore updated vertex duals are still multiples of 0.5.
• Blossom duals increase or decrease by 2*δ, therefore updated blossom duals are still integers.

The value of vertex dual variables and blossom dual variables never exceeds the greatest edge weight in the graph. This may be helpful for choosing an integer data type for the dual variables. (Alternatively, choose a programming language with unlimited integer range. This is perhaps the thing I love most about Python.)

Proof that dual variables do not exceed max-weight:

• Vertex dual variables start at u_x = 0.5*max-weight.
• While the algorithm runs, there is at least one vertex which has been unmatched since the beginning. This vertex has always had label S, therefore its dual always decreased by δ during a dual variable update. Since it started at 0.5*max-weight and can not become negative, the sum of δ over all dual variable updates can not exceed 0.5*max-weight.
• Vertex dual variables increase by at most δ per update. Therefore no vertex dual can increase by more than 0.5*max-weight in total. Therefore no vertex dual can exceed max-weight.
• Blossom dual variables start at z_B = 0.
• Blossom dual variables increase by at most 2*δ per update. Therefore no blossom dual can increase by more than max-weight in total. Therefore no blossom dual can exceed max-weight.

Handling floating point edge weights

Floating point calculations are subject to rounding errors. This has two consequences for the matching algorithm:

• The algorithm may return a matching which has slightly lower weight than the actual maximum weight.

• The algorithm may not reliably recognize tight edges. To check whether an edge is tight, its slack is compared to zero. Rounding errors may cause the slack to appear positive even when an exact calculation would show it to be zero. The slack of some edges may even become slightly negative.

  I believe this does not affect the correctness of the algorithm. An edge that should be tight but is not recognized as tight due to rounding errors, can be pulled tight through an additional dual variable update. As side-effect of this update, the edge will immediately be used to grow the alternating tree, or construct a blossom or augmenting path. This mechanism allows the algorithm to make progress, even if slack comparisons are repeatedly thrown off by rounding errors. Rounding errors may cause the algorithm to perform more dual variable updates than strictly necessary. But this will still not cause the run time of the algorithm to exceed O(n³).

It seems to me that the matching algorithm is stable for floating point weights. And it seems to me that it returns a matching which is close to optimal, and could have been optimal if edge weights were changed by small amounts.

I must admit these arguments are mostly based on intuition. Unfortunately I don't know how to properly analyze the floating point accuracy of this algorithm.

Finding a maximum weight matching out of all maximum cardinality matchings

It is sometimes useful to find a maximum cardinality matching which has maximum weight out of all maximum cardinality matchings. A simple way to achieve this is to increase the weight of all edges by the same amount.

In general, a maximum weight matching is not necessarily a maximum cardinality matching. However if all edge weights are at least n times the difference between the maximum and minimum edge weight, any maximum weight matching is guaranteed to have maximum cardinality. Proof: The weight of a non-maximum-cardinality matching can be increased by matching an additional edge. In order to match that extra edge, some high-weight edges may have to be removed from the matching. Their place might be taken low-weight edges. But even if all previously matched edges had maximum weight, and all newly matched edges have minimum weight, the weight of the matching will still increase.

Changing all edge weights by the same amount does not change the relative preference for a certain groups of edges over another group of the same size. Therefore, given that we only consider maximum-cardinality matchings, changing all weights by the same amount doesn't change which of these matchings has maximum weight.

References

1. Jack Edmonds, "Paths, trees, and flowers." Canadian Journal of Mathematics vol. 17 no. 3, 1965. (link) (pdf)

2. Jack Edmonds, "Maximum matching and a polyhedron with 0,1-vertices." Journal of research of the National Bureau of Standards vol. 69B, 1965. (pdf)

3. Harold N. Gabow, "Implementation of algorithms for maximum matching on nonbipartite graphs." Ph.D. thesis, Stanford University, 1974.

4. Z. Galil, S. Micali, H. Gabow, "An O(EV log V) algorithm for finding a maximal weighted matching in general graphs." SIAM Journal on Computing vol. 15 no. 1, 1986. (link) (pdf)

5. Zvi Galil, "Efficient algorithms for finding maximum matching in graphs." ACM Computing Surveys vol. 18 no. 1, 1986. (link) (pdf)

6. Harold N. Gabow, "Data structures for weighted matching and nearest common ancestors with linking." Proc. 1st ACM-SIAM symposium on discrete algorithms, 1990. (link) (pdf)

---

Written in 2023 by Joris van Rantwijk. This work is licensed under CC BY-ND 4.0.