Attributed Tree Matching and Maximum Weight Cliques

A classical way of matching relational structures consists of finding a maximum clique in a derived “association graph.” However, it is not clear how to apply this approach to problems where the graphs are hierarchically organized, i.e. are trees, since maximum cliques are not constrained to preserve the partial order. We have recently provided a solution to this problem by constructing the association graph using the graph-theoretic concept of connectivity. In this paper, we extend the approach to the problem of matching attributed trees. Specifically, we show how to derive a “weighted” association graph, and prove that the attributed tree matching problem is equivalent to finding a maximum weight clique in it. We then formulate the maximum weight clique problem in terms of a continuous optimization problem, which we solve using “replicator” dynamical systems developed in theoretical biology. This formulation is attractive because it can motivate analog and biological implementations. We illustrate the power of the approach by matching articulated and deformed shapes described by shock trees.