Objective of Hungarian method

Given the above introductory definitions regarding bipartite graphs, the objective of the Hungarian algorithm is to find the maximum weight matching M in a complete bipartite graph,G(X, Y, E).

Two fundamental notions that allow us to significantly reduce the search space of perfect matchings M to find the maximum weight matching are the concepts ofvertex labeling andequality graphs.

DEFINITION 4.7.– (Feasible vertex labeling): a feasible vertex labeling upon a weighted bipartite graph G(X, Y, E) is a function that assigns a label, l ∈ Z to each vertex in V, such that the sum of labels of any pair of vertices connected by an edgeex,y is greater than equal to the edge weightwxy,

l:V → Z | l(x) +l(y)≥wxy, ∀x∈X, y ∈X

DEFINITION 4.8.– (Equality (subgraph):) given a feasible vertex labeling,l, the equality subgraph of a complete bipartite graph is the graphGl(X, Y, El)defined by the vertices inXand

Y and a subset of edgesEl ∈E, whose weightswxy are strictly equal to the sum of vertex labelsl(x) +l(y):

El={(x, y)} | l(x) +l(y) =wxy

Given the vertex labeling and equality subgraph definitions, the basis of the Hungarian algorithm is the Huhn – Monkres theorem.

Theorem 4.1. (Huhn – Monkres theorem): if a perfect matching is found in an equality subgraph ofG(X, Y, E), this matching is the maximum weight matching.

Proof 1. From the definition of feasible labeling, any edge (x, y)∈E satisfies that

w(x, y)≤l(x) +l(y) [4.6]

For a perfect matching M, each vertex is only adjacent to one edge, thus,

Now, any edgeex,yin an equality graph satisfies

w(x, y) =l(x) +l(y) [4.8]

Hence, for any perfect matching over the equality graph,Ml, it

yields

Finally, by merging equations [4.7] and [4.10]

Thus, the problem of finding a maximum weighted matching is transformed into the onefinding a feasible vertex labeling with a perfect matching in the associated equality subgraph. This is essentially achieved by selecting an initial vertex labeling as well as a matchingM, of size |M|, in the equality graph, and iteratively growingM until it becomes a perfect matching (|M|=k). In each iteration, the size of M is increased by one edge after anaugmented pathis found.

DEFINITION 4.9.– (Path): a path over a graph G(V, E) is defined as a sequence of vertices{v¹, v², v³, v⁴, . . . , v^p}, such that there exists an edge connecting each pair of vertices (vⁱ, vⁱ⁺¹). Note that the superscripts 1,2, . . . , p are not referred to the indices of the vertices in the graph, but to their order in the path sequence.

DEFINITION 4.10.– (Augmented path): given a matchingM in the equality graph, an augmented path is a path (1) whose edges alternate between M and M¯ (alternating path), and (2) whose start and end vertices, v¹ and v^p, are unmatched, i.e., {(v¹, v²)∈M¯,(v², v³)∈M,(v³, v⁴)∈M, . . . ,¯ (vⁱ, vⁱ⁺¹)∈M}¯ .

Obviously, if an augmented path is found, the size of M can be increased by one edge by inverting the edges in the path from M¯ → M and M → M¯ so that the new path can be expressed as (Figure 4.4(a)) {(v¹, v²) ∈ M, (v², v³) ∈ M, . . . ,¯ (vⁱ, vⁱ⁺¹)∈M}(Figure 4.4(b)).

As mentioned earlier, the Hungarian algorithm starts by an arbitrary vertex labeling. Typically, the labels for the vertices inY are set to 0, while each vertexxi ∈Xis labeled with the maximum of its incident edges,

l(yi) = 0 [4.11]

l(xi) = max

y_i∈Y(w(xi, yi)) [4.12]

x₁ M y₁

x₃ y₃

x₂ y₂

(a)

x₁

x₃ x₂

y₁

y₃ y₂ Augmenting path

(b) x₁

x₃ x₂

y₁

y₃ y₂ M

(b)

Figure 4.4.Illustration of augmented paths. (a) Bipartite graph with two disjoint subsets and an arbitrary matchingM; (b) Alternating path with start aty1and end atx3. The vertex order is

indicated by the arrows. (c) The alternating path allows us to increase the matchingM, resulting in a perfect matching

Then, a matching in the equality graph,El, associated with the vertex labeling is selected. If |M| = k, this matching is already perfect and the optimum is found. If the matching is not perfect, the size of M needs to be gradually incremented in a number of iterations. By definition 4.10, |M| can be increased by one edge if an augmenting path is found. Thus, each iteration step is directed toward the search for an augmented path.

As M is not yet perfect, there must be some unmatched vertex x ∈ X connected to a matched vertex y. This seems

to be obvious, since otherwise the vertexxi would be already matched in M. Let Nl(x) denote the subset of vertices in Y which are connected to x (the neighbors of x). Also, let X denote the subset of vertices X ∈ X − {x} matched to any vertex inNl(x). Thus,xis “competing” withX for an edge in M. The path {x, y, x ∈ X} can be thought of as a section of an augmented path. Now, if an unmatched vertexy is found, connected to any vertex inS = {X ∪xi} in a (new) equality graph, two situations may occur:

1.y is connected tox. Then, M can be increased by adding a new edge (x, y). The new matching can be expressed as M =M ∪(x, y),

2.y is connected to a vertex in X. Let this vertex be denoted as x2, and y2, the vertex in Y matched to x2. Then, an augmented path can be found in the formx, y2, x2, y, and M can be incremented by inverting the path edges fromM to M¯ and vice versa.

Now, assuming that a maximum size matchingMhas been previously selected, the required unmatched vertex y does not yet exist in El. Otherwise, this vertex would already be included in M. Therefore, the equality graph El must be expanded to find new potential vertices in Y to augment M. Obviously, the expansion ofElrequires a vertex (re)labelingl. It is formulated as

δl = min

x∈S,y∈N_l(S)(l(x) +l(y)−w(x, y)) [4.13]

l(v) =

⎧⎨

⎩

l(v)−δl, v∈S l(v) +δl, v∈Nl(S)

l(v), otherwise [4.14]

The relabeling function l ensures that a new equality graph is found, E_l, such that E_l ∈ El and some new edge (xi ∈ S, y /∈ Nl(S)) exists. In other words, the new set of S neighbors inE_l isN_l(S) =Nl(S)∪ {y}.

Consider the new edges (x ∈ S, y /∈ Nl(S)) in E_l. With reference to the vertex y, two possible cases are to be considered:

–xi : (xi, y)∈ M (y is not matched). Thus, an augmented path can be found and|M|=|M|+ 1.

–∃xi : (xi, y) ∈ M (y is matched). The new edge can be expressed as (xi ∈ S, y /∈ Nl(S) ∈ M). Since y is not an unmatched vertex, the path cannot be augmented. In this case, the vertex x matched to y is attached to S so that y belongs to N_l(S). Then, a new vertex relabeling is required, forcing new edges in E_l connecting vertices from S toY −N_l(S). Such vertex labeling is iterated, adding vertices toS andNl(S)until an unmatched vertex inY is found.

Each time |M| is incremented, the previous steps are repeated, starting with another free vertex inX. This process is iterated until all free vertices in X are explored, in which case |M| = k and a perfect matching (the maximum weight matching) is achieved.