Foundations of Data Science

graph. The vertices of Π are denoted by dotted circles in the above tree.
Consider a set of trees where each tree has a root that corresponds to one of the constraints.
If the constraint at each root is satisfied by the edge of the MWM, then we have
found the MWM. Suppose that the matching at the root in one of the trees disagrees
with the MWM. Then there is an alternating path of vertices of length 2k consisting of
vertices corresponding to edges in Π and edges in Λ. Map this path onto the bipartite
graph. In the bipartite graph the path will consist of a number of cycles plus a simple
path. If k is large enough there will be a large number of cycles since no cycle can be of
length more than 2n. Let m be the number of cycles. Then m ≥ 2k
2n = k n .
Let π ∗ be the MWM in the bipartite graph. Take one of the cycles and use it as an
alternating path to convert the MWM to another matching. Assuming that the MWM
is unique and that the next closest matching is ε less, W π∗ − W π > ε where π is the new
matching.
Consider the tree matching. Modify the tree matching by using the alternating path
of all cycles and the left over simple path. The simple path is converted to a cycle by
adding two edges. The cost of the two edges is at most 2w* where w* is the weight of the
maximum weight edge. Each time we modify Λ by an alternating cycle, we increase the
cost of the matching by at least ε. When we modify Λ by the left over simple path, we
increase the cost of the tree matching by ε − 2w∗ since the two edges that were used to
create a cycle in the bipartite graph are not used. Thus
weight of Λ - weight of Λ ′ ≥ k n ε − 2w∗
which must be negative since Λ ′ is optimal for the tree. However, if k is large enough
this becomes positive, an impossibility since Λ ′ is the best possible. Since we have a
tree, there can be no cycles, as messages are passed up the tree, each sub tree is optimal
and hence the total tree is optimal. Thus the message passing algorithm must find the
maximum weight matching in the weighted complete bipartite graph assuming that the
maximum weight matching is unique. Note that applying one of the cycles that makes
up the alternating path decreased the bipartite graph match but increases the value of
the tree. However, it does not give a higher tree matching, which is not possible since
we already have the maximum tree matching. The reason for this is that the application
of a single cycle does not result in a valid tree matching. One must apply the entire
alternating path to go from one matching to another.
9.12 Warning Propagation
Significant progress has been made using methods similar to belief propagation in
finding satisfying assignments for 3-CNF formulas. Thus, we include a section on a
version of belief propagation, called warning propagation, that is quite effective in finding
assignments. Consider a factor graph for a SAT problem. Index the variables by i, j, and
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