Foundations of Data Science

9.11 Maximum Weight Matching
We have seen that the belief propagation algorithm converges to the correct solution
in trees and graphs with a single cycle. It also correctly converges for a number of problems.
Here we give one example, the maximum weight matching problem where there is
a unique solution.
We apply the belief propagation algorithm to find the maximal weight matching
(MWM) in a complete bipartite graph. If the MWM in the bipartite graph is unique,
then the belief propagation algorithm will converge to it.
Let G = (V 1 , V 2 , E) be a complete bipartite graph where V 1 = {a 1 , . . . , a n } , V 2 =
{b 1 , . . . , b n } , and (a i , b j ) ∈ E, 1 ≤ i, j ≤ n. Let π = {π (1) , . . . , π (n)} be a permutation
of {1, . . . , n}. The collection of edges {( ) ( )}
a 1 , b π(1) , . . . , an , b π(n) is called a
matching which is denoted by π. Let w ij be the weight associated with the edge (a i , b j ).
∑
The weight of the matching π is w π = n w iπ(i) . The maximum weight matching π ∗ is
π ∗ = arg max
π
w π
i=1
The first step is to create a factor graph corresponding to the MWM problem. Each
edge of the bipartite graph is represented by a variable c ij which takes on the value zero or
one. The value one means that the edge is present in the matching, the value zero means
that the edge is not present in the matching. A set of constraints is used to force the set
of edges to be a matching. The constraints are of the form ∑ c ij = 1 and ∑ c ij = 1. Any
j
i
0,1 assignment to the variables c ij that satisfies all of the constraints defines a matching.
In addition, we have constraints for the weights of the edges.
We now construct a factor graph, a portion of which is shown in Fig. 9.8. Associated
with the factor graph is a function f (c 11 , c 12 , . . .) consisting of a set of terms for each c ij
enforcing the constraints and summing the weights of the edges of the matching. The
terms for c 12 are
( ) ∣( ) ∑ ∣∣∣∣ ∑ −λ
c i2 − 1
∣
∣ − λ c 1j − 1
∣ + w 12c 12
i
j
where λ is a large positive number used to enforce the constraints when we maximize the
function. Finding the values of c 11 , c 12 , . . . that maximize f finds the maximum weighted
matching for the bipartite graph.
If the factor graph was a tree, then the message from a variable node x to its parent
is a message g(x) that gives the maximum value for the sub tree for each value of x. To
compute g(x), one sums all messages into the node x. For a constraint node, one sums
all messages from sub trees and maximizes the sum over all variables except the variable
of the parent node subject to the constraint. The message from a variable x consists of
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