Matching Free Trees, Maximal Cliques, and Monotone Game Dynamics

Abstract

Abstract-Motivated by our recent work on rooted tree matching, in this paper we provide a solution to the problem of matching two free (i.e., unrooted) trees by constructing an association graph whose maximal cliques are in one-to-one correspondence with maximal common subtrees. We then solve the problem using simple payoff-monotonic dynamics from evolutionary game theory. We illustrate the power of the approach by matching articulated and deformed shapes described by shape-axis trees. Experiments on hundreds of larger, uniformly random trees are also presented. The results are impressive: despite the inherent inability of these simple dynamics to escape from local optima, they always returned a globally optimal solution.

Payoff-Monotonic Game Dynamics and the Maximum Clique Problem

“…Note that model A.1 is the standard discrete-time replicator dynamics, which have already proven to be remarkably effective in tackling maximum clique and related problems, and to be competitive to other more elaborated neural network heuristics (Bomze, 1997;Bomze et al, 1997Bomze et al, , 2000Pelillo, 1995Pelillo, , 1999Pelillo et al, 1999). Equation A.2 has been used in Pelillo (1999Pelillo ( , 2002 as a heuristic for graph and tree isomorphism problems.…”

“…In a recent series of papers (Pelillo, 1995(Pelillo, , 1999(Pelillo, , 2002Bomze, 1997;Bomze, Pelillo, & Giacomini, 1997;Bomze, Pelillo, & Stix, 2000;Jagota et al, 2000;Bomze, Budinich, Pelillo, & Rossi, 2002;Pelillo, Siddiqi, & Zucker, 1999), a completely different framework has been developed. The approach is centered around a classic result from graph theory due to Motzkin and Straus (1965), and variations thereof, which allow us to formulate the MCP as a standard quadratic program-namely, a continuous quadratic optimization problem with simplex (or probability) constraints, to solve which replicator equations have been remarkably effective despite their simplicity.…”

Enhancement of incremental design for FPGAs using circuit similarity

“…The predefined similarity is used to capture non-topological connections between two graphs. The algorithm terminates when the difference between of the total similarity scores in two consecutive iterations is smaller [15] Identifying a bijection between the nodes of two graphs which preserves (directed) adjacency.…”

Annealed imitation: fast dynamics for maximum clique

“…The MotzkinStraus formulation, and variations thereof, has motivated various neural network heuristics for maximum clique. In particular, replicator equations from evolutionary game theory have proven to be quite effective in solving this and related combinatorial optimization problems [1], [4], [12], [13], [15].…”

“…The following result, proved in [7], [13], generalizes the celebrated fundamental theorem of natural selection [8], [16].…”