{"title": "Matching Free Trees with Replicator Equations", "book": "Advances in Neural Information Processing Systems", "page_first": 865, "page_last": 872, "abstract": null, "full_text": "Matching Free Trees with Replicator Equations\n\nMarcello Pelillo\n\nDipartimento di Informatica\n\nUniversit`a Ca\u2019 Foscari di Venezia\n\nVia Torino 155, 30172 Venezia Mestre, Italy\n\nE-mail: pelillo@dsi.unive.it\n\nAbstract\n\nMotivated by our recent work on rooted tree matching, in this paper we\nprovide a solution to the problem of matching two free (i.e., unrooted)\ntrees by constructing an association graph whose maximal cliques are\nin one-to-one correspondence with maximal common subtrees. We then\nsolve the problem using simple replicator dynamics from evolutionary\ngame theory. Experiments on hundreds of uniformly random trees are\npresented. The results are impressive: despite the inherent inability of\nthese simple dynamics to escape from local optima, they always returned\na globally optimal solution.\n\n1 Introduction\n\nGraph matching is a classic problem in computer vision and pattern recognition, instances\nof which arise in areas as diverse as object recognition, motion and stereo analysis [1]. In\nmany problems (e.g., [2, 11, 19]) the graphs at hand have a peculiar structure: they are\nconnected and acyclic, i.e.\nthey are free trees. Note that, unlike \u201crooted\u201d trees, in free\ntrees there is no distinguished node playing the role of the root, and hence no hierarchy is\nimposed on them. Standard graph matching techniques, such as [8], yield solutions that are\nnot constrained to preserve connectedness and hence cannot be applied to free trees.\n\nA classic approach to solving the graph matching problem consists of transforming it into\nthe equivalent problem of \ufb01nding a maximum clique in an auxiliary graph structure, known\nas the association graph [1]. This framework is attractive because it casts graph matching\nas a pure graph-theoretic problem, for which a solid theory and powerful algorithms have\n-\nbeen developed. Note that, although the maximum clique problem is known to be \u0002\u0001\nhard, powerful heuristics exist which ef\ufb01ciently \ufb01nd good approximate solutions [4].\n\nMotivated by our recent work on rooted tree matching [15], in this paper we propose a\nsolution to the free tree matching problem by providing a straightforward way of deriv-\ning an association graph from two free trees. We prove that in the new formulation there\nis a one-to-one correspondence between maximal (maximum) cliques in the derived asso-\nciation graph and maximal (maximum) subtree isomorphisms. As an obvious corollary,\nthe computational complexity of \ufb01nding a maximum clique in such graphs is therefore the\nsame as the subtree isomorphism problem, which is known to be polynomial in the number\nof nodes [7].\n\nFollowing [13, 15], we use a recent generalization of the Motzkin-Straus theorem [12] to\n\n\fformulate the maximum clique problem as a quadratic programming problem. To (approxi-\nmately) solve it we employ replicator equations, a class of simple continuous- and discrete-\ntime dynamical systems developed and studied in evolutionary game theory [10, 17].\n\nWe illustrate the power of the approach via experiments on hundreds of (uniformly) random\ntrees. The results are impressive: despite the counter-intuitive maximum clique formulation\nof the tree matching problem, and the inherent inability of these simple dynamics to escape\nfrom local optima, they always found a globally optimal solution.\n\n2 Subtree isomorphisms and maximal cliques\n\n\u001f\"!\n)+*-,.\u0003/\r0\u000b\n\r012\r435(2(-(6\r87\n\n. If\n\nbe a graph, where\n\nLet\nedges. The order of\nTwo nodes\nedge. The adjacency matrix of\n\n\u0002\u0001\u0004\u0003\u0006\u0005\b\u0007\n\t\f\u000b\n\nis the number of nodes in\n\nare said to be adjacent (denoted\n\nis the set of nodes and\n\nis the set of (undirected)\n, while its size is the number of edges.\n) if they are connected by an\n\nis the\n\nsymmetric matrix\n\nde\ufb01ned as\n\n\u000e\u0007\n\u000f\u0011\u0010\u0012\u0005\n\n\u0014\u0013\u0015\u000f\n\n\u0016\u0018\u0017\u0019\u0016\n$\u0004%\n\nif\notherwise\n\n\u0013&\u000f\n\n\u001a\u001c\u001b\u0014\u0001\u001d\u0003\u0006\u001e \u001f\"!#\u000b\n\n, denoted\n\nThe degree of a node\nany sequence of distinct nodes\ncase, the length of the path is\nto be connected if any two nodes are joined by a path. The distance between two nodes\nand\n\n, is the number of nodes adjacent to it. A path is\n; in this\nthe path is called a cycle. A graph is said\n\n, is the length of the shortest path joining them (by convention\n, the induced\n\n, if there is no such path). Given a subset of nodes\n\n01@\u0013A\rB7\n\nsuch that for all\n\n, denoted by\n\n3>\u0013?\n\n(2(2(;\u0016\n\nC8\u0003/\r5\u0007;\u000f+\u000bD\u0001FE\nLK\n\nis the graph having\n\nsubgraph\nif and only if they are adjacent in\n. A connected graph with no cycles is called a free tree,\nor simply a tree. Trees have a number of interesting properties. One which turns out to\nbe very useful for our characterization is that in a tree any two nodes are connected by a\nunique path.\n\nas its node set, and two nodes are adjacent in\n\nGIHJ\u0005\n\nC8\u0003/\r5\u0007;\u000f+\u000b\n\nLK\n\nGNM\n\nGNM\n\n9:\u0001\n\n\u001f=<\n\n,\n\n\u0007P\t\n\nLet\n\nand\n\nand\n\n3[Z\n\nH?\u0005\n\n\u0001J\u0003\u0006\u0005\n\nif and only if\n\n, we have\nand\n\nX[3\u001cH]\u000583\n\nbe two trees. Any bijection\n\nO0QR\u0001J\u0003\u0006\u0005.QS\u0007\n\tTQ#\u000b\n\r5\u0007;\u000f[\u0010^X_3\nXR3\n\n, with\n, is called a subtree isomorphism if it preserves both the adjacency\nrelationships between the nodes and the connectedness of the matched subgraphs. For-\nmally, this means that, given\nand, in\nare connected. A subtree isomorphism\naddition, the induced subgraphs\nis maximal if there is no other subtree isomorphism\na strict sub-\nhas largest cardinality. The maximal (maximum) subtree\nset of\nisomorphism problem is to \ufb01nd a maximal (maximum) subtree isomorphism between two\ntrees. A word of caution about terminology is in order here. Despite name similarity, we\nare not addressing the so-called subtree isomorphism problem, which consists of determin-\ning whether a given tree is isomorphic to a subtree of a larger one. In fact, we are dealing\nwith a generalization thereof, the maximum common subtree problem, which consists of\ndetermining the largest isomorphic subtrees of two given trees. We shall continue to use\nour own terminology, however, as it emphasizes the role of the isomorphism\n\nUWVYX\nX\\Q\nU5\u0003/\r0\u000b\b\u0013]U\u000e\u0003=\u000f \u000b\n\n\u0019\u0013W\u000f\nXaQ2M\n\n, and maximum if\n\nU4bcV0X^b\n\nO0Q`K\n\nX\u0011b\n\nX^b\n\nXR3\n\nwith\n\n.\n\nThe free tree association graph (FTAG) of two trees\nthe graph\n\nwhere\n\nand\n\nOd3\u001c\u0001e\u0003\u0006\u0005B3f\u0007P\tN3-\u000b\n\n\u0001g\u0003h\u0005\n\n\u0007\n\t\n\ni\u0001\u0015\u0003\u0006\u0005\b\u0007\n\tj\u000b\n\nand, for any two nodes\n\nand\n\n\u0005\u001d\u0001W\u0005\n\nin\n\n\u0017@\u0005.Q\n\n, we have\n\n\u0003/\u000f8\u0007\nl \u000b\n\n\u0003/\r5\u0007;kN\u000b\n\u0003=\r\u000e\u0007\nkN\u000b\b\u0013\u001d\u0003/\u000f8\u0007\nl \u000bnmoC.\u0003=\r\u000e\u0007\n\u000f \u000bp\u0001WC8\u0003/kq\u0007Plr\u000b\b(\n\nNote that this de\ufb01nition of the association graph is stronger than the standard one used for\nmatching arbitrary relational structures [1].\n\nis\n\n(1)\n\n(2)\n\n\u0005\n\t\n\n\u0005\n\n\u001e\n\u0001\n\u0007\n\u000f\n\u001f\n!\n'\n\u0007\n(\n\n%\n\n\u001f\n\u0016\n\n\u000f\nG\n\nO\n3\n3\n3\n\u000b\nX\nQ\nQ\nO\n3\nK\nX\n3\nM\n3\nZ\nQ\n3\nU\nO\nQ\nQ\nQ\n\u000b\n3\n\u0005\n\fis said to be a clique if all its nodes are mutually adjacent. A\nA subset of vertices of\nmaximal clique is one which is not contained in any larger clique, while a maximum clique\nis a clique having largest cardinality. The maximum clique problem is to \ufb01nd a maximum\nclique of\n\n.\n\nThe following theorem, which is the basis of the work reported here, establishes a one-to-\none correspondence between the maximum subtree isomorphism problem and the maxi-\nmum clique problem.\n\nTheorem 1 Any maximal (maximum) subtree isomorphism between two trees induces a\nmaximal (maximum) clique in the corresponding FTAG, and vice versa.\n\nProof (outline). Let\nand\n\n, and let\n\n\u0002\u0001\nU5\u0003/\r0\u000b;\u000b>Vr\n\n\u0001\u0003\u0002r\u0003=\r\u000e\u0007\n\nThis clearly implies that\nclique. Trivially,\npart of the theorem.\n\nbe a maximal subtree isomorphism between trees\n\nU&VYX_3\n\u0003h\u0005:\u0007P\t\f\u000b\n\u0010\u0018X\n3\u0005\u0004\nC.\u0003=\r\u000e\u0007\n\u000f \u000b\\\u0001\u0002C.\u0003hU\u000e\u0003=\rB\u000b\n\ndenote the corresponding FTAG. Let\nbe de\ufb01ned as\n. From the de\ufb01nition of a subtree isomorphism it follows that\n.\nis a\nis maximal, and this proves the \ufb01rst\n\nonto the path joining\n\n, and therefore\n\nU5\u0003=\rB\u000b\n\nis a maximal clique because\n\n\u000e\u0007\n\u000f\u0018\u0010\n\u0007PU5\u0003/\u000f+\u000b;\u000b\n\nOn3\nU\u000e\u0003=\u000f \u000b\n\nfor all\n\n?\u0010\n\nX_3\n\nand\n\nmaps the path between any two nodes\n\nSuppose now that\n\nand\n\nG\u0015\u0001\u0003\u0002r\u0003=\n\n\u0007;k\n\u0007\u0007\u0006\b\u0006\u0007\u00064\u0007#\u0003/\r\n\u0001\t\u0002-k>3f\u0007\u0007\u0006\u0007\u0006\b\u00060\u0007;k\n\n\u0007;k\n\n\u0002-\r43#\u0007\b\u0006\u0007\u0006\u0007\u0006Y\u0007;\rB7\n9d\u0001\n\n. From the de\ufb01nition of the FTAG and the hypothesis that\n\nfor all\nis simple to see that\ntrivially preserves the adjacency relationships between nodes. The fact that\nisomorphism is a straightforward consequence of the maximality of\n\nis a one-to-one and onto correspondence between\n\nH&\u0005B3\n(-(2(;\u0016\n\nU\u0011VrXR3\n\nHW\u0005\n\n.\n\nand\n\n,\nis a clique, it\n, which\nis a maximal\n\n\u000bp\u0001Wk\n\nU5\u0003/\r\nXaQ\n\nis a maximal clique of\n. De\ufb01ne\n\n, and let\nas\n\nTo conclude the proof we have to show that the subgraphs that we obtain when we restrict\nourselves to\n, are trees, and this is equivalent to showing\nthat they are connected. Suppose by contradiction that this is not the case, and let\n\n, i.e.\n\nand\n\nand\n\nbe two nodes which are not joined by a path in\n\n, however, there must exist a path\n\nof\n\nX[3\nOY3\n\n\u0011\u0010\f\u0001\u0012\n\u0011\u0013\n\u0010a\u0001\n\n, for some\nbe the\n\n(remember that\n\n-th node on the path\n\n(2(-(\r\u0015\n\u000bN\u0001\u0015C8\u0003/k\n\n\u0007\n\n\u0007;k\n\n, be a node on this path which is not in\n\nO0Q\n\nXaQ2M\n\u0001\u000b\n\n, and hence\n\njoining them in\n\n\u0007\n\nand\n\nare nodes\n. Let\n. Moreover, let\nin\n). It is easy to show\nis\nis a node on\n\nO\u000e3\n\nand\n\nwhich joins\n\n. Since both\n\nOn3\n1\f\nB3n(-(2(\r\n\u000f\u000e\u0002\u0001g\r\n3n(2(2(\n\u0007;k\n\nXL3\n\u000eA\u0001\nC.\u0003=k\nC8\u0003/\r5\u0007\u001b\n8\u000b\u001d\u001c\u0014C8\u0003\u001e\n\u000e\u0007;\u000f+\u000b\n\n\u000b>\u0001\u0003\u0015\n\nC.\u0003=\r\u000e\u0007\n\u000f \u000b:\u0001\n\nthat the set\na maximal clique. This can be proved by exploiting the obvious fact that if\nthe path joining any two nodes\n\nis a clique, thereby contradicting the hypothesis that\n\n\u0002r\u0003\u0017\n\u0018\u0010\n\n, then\n\nand\n\n.\n\nThe \u201cmaximum\u201d part of the statement is proved similarly.\n\nXaQ\n\n\u0014\u0019\u0001\nC8\u0003/\r\n\u00102\u000b\n\u0004\u001a\u0019\n\nThe FTAG is readily derived by using a classical representation for graphs, i.e., the so-\ncalled distance matrix which, for an arbitrary graph\nmatrix\n. Ef\ufb01cient,\nclassical algorithms are available for obtaining such a matrix [6]. Note also that the distance\nmatrix of a graph can easily be constructed from its adjacency matrix\n. In fact, denoting\nby\nequals\nthe least\n\n, we have that\nsince a tree is connected).\n\n, the distance between nodes\n\n(there must be such an\n\n-th entry of the matrix\n\n\u0001\u0002\u0003h\u0005:\u0007P\t\f\u000b\n\n-th power of\n\n\u0001?C8\u0003/\n\n\u0016\u0014\u0017^\u0016\n\nfor which\n\nof order\n\n, is the\n\nwhere\n\n\u001a\u001c\u001b\n\n, the\n\nand\n\nthe\n\n\u0007\n\n\u001f\"!\n\n\u001f\"!\n\n\u0003=C\n\u0007! \u000b\n\n\u0003/9\n\n3 Matching free trees with replicator dynamics\n\n!#\"\n\nLet\n\n\u001d\u0001\nIR7 :\n\n\u0003\u0006\u0005\b\u0007\n\tj\u000b\n\nbe an arbitrary graph of order\n\n, and let\n\n$07L\u0001%\u0002\u001a&\n\nIR7\n\nV('\n\n&^\u0001\n\n% and\n\n\u001f*)\n\n$Y7\n\u0007Y9d\u0001\n\n(2(2(;\u0016\n\ndenote the standard simplex of\n\n\n\nZ\nX\nQ\nO\nQ\nG\n\u0001\nH\n\u0005\nG\n\u0001\nU\nX\n3\nG\n\u0001\nG\n\u0001\nU\n3\n3\n\u000b\n7\n7\n\u000b\n\u0004\n\nX\n3\n\u0001\n\u0004\nX\nQ\n7\n\u0004\nQ\nZ\nX\nQ\n\u001f\n\u001f\n%\nG\nU\nX\n3\nU\nG\nX\n3\nO\n3\nK\nX\n3\nM\nK\n\n\u001f\n!\n\u0010\nK\nM\n\n\u001f\n\n!\n\n\u001f\n!\n%\nX\n3\n\u0016\n\u0016\n\u0013\n\u0014\nk\n\u001f\n\u0001\n\u0016\n1\n\u0016\n\u0016\nk\n!\nk\n\u001f\nk\n!\nO\nQ\n\u001f\n!\n\u001f\n!\n\u000b\n\u001f\n!\n\u0007\n\u0016\nG\nH\n\u0005\nG\n\n\u000f\n\n\u0016\n\u001f\n\u0001\n\u000b\nC\n\u001f\n!\n\u000b\n\n\u001f\n\n!\n\u001e\n7\n\u001f\n!\n\u001a\n7\n\u001b\n\u0016\n\u001a\n\u001b\nC\n\u001f\n!\n\u0016\n\u001e\n7\n\u001f\n'\n\u0016\n\u0016\n\u0010\nb\n\n'\n%\n\u0004\n\fwhere\na subset of vertices\nin\n\nde\ufb01ned as\n\nis the vector whose components equal 1, and a prime denotes transposition. Given\nits characteristic vector which is the point\n\n, we will denote by\n\nof\n\nwhere\u0004\n\n\u0004 denotes the cardinality of\n\nNow, consider the following quadratic function\n\nif\notherwise\n\n9p\u0010@G\n\n(3)\n\n&\u0001\n\n%\u0003\u0002\u0005\u0004\n\n.\n\n\u0003\u0017&Y\u000b\b\u0001\n\n. The following theorem, recently proved\nwhere\nby Bomze [3], expands on the Motzkin-Straus theorem [12], a remarkable result which es-\ntablishes a connection between the maximum clique problem and quadratic programming.\n\nis the adjacency matrix of\n\n\u0001g\u0003=\u001e\n\nin\n\nin\n\nin\n\nif and only if\n\n.\n\non\n\n$07\n\nthe other hand.\n\nTheorem 2 Let\nvector. Then,\n\nbe a subset of vertices of a graph\n\n, and let\n\nis a maximal (maximum) clique of\n\nis a local (global)\nare\n\nstrict and are characteristic vectors of maximal cliques of\n\nUnlike the original Motzkin-Straus formulation, which is plagued by the presence of \u201cspu-\n\nWe now turn our attention to a class of simple dynamical systems that we use for solving\n\nare strict, and are characteristic vectors of maximal/maximum cliques in\n. In a formal\nsense, therefore, a one-to-one correspondence exists between maximal cliques and local\non the one hand, and maximum cliques and global maximizers on\n\n&\b be its characteristic\n&\t\n. Moreover, all local (and hence global) maximizers of\u0006\nmaximizer\u0006\nrious\u201d solutions [14], the previous result guarantees us that all maximizers of\u0006\nmaximizers of\u0006\nour quadratic optimization problem. Let\n\n\u000f\u0011\u0010\n\n.\u001fP\u0003\r\f;\u000b\b\u0001\n\n.\u001fP\u0003\r\f;\u000b\n\n8\u001f\n\u0003\u0012\f\n\nwhere a dot signi\ufb01es derivative with respect to time, and its discrete-time counterpart:\n\nmatrix, and consider the following continuous-time dynamical system:\n\nbe a non-negative real-valued\n\n\u001f\n\u0003\u0012\f;\u000b\b\u0013\n!\u0016\u0015\n\n8\u001f\n\u0003\u0012\f;\u000b\n!\u0016\u0015\n\n!r\u0003\r\f;\u000b\u0018\u0017\n\n\u0017@\u0016\n\n(4)\n\n\u0001A\u0003/kN\u001f\"!\n\nwhere\n\n(5)\n\n(6)\n\nBoth (4) and (5) are called replicator equations in evolutionary game theory, since they\nare used to model evolution over time of relative frequencies of interacting, self-replicating\nentities [10, 17]. It is readily seen that the simplex\nis invariant under these dynamics,\nwhich means that every trajectory starting in\nfor all future times, and\ntheir stationary points coincide.\n\nwill remain in\n\n$\u000e7\n\n$47\n\nWe are now interested in the dynamical properties of replicator dynamics; it is these prop-\nerties that will allow us to solve our original tree matching problem. The following result\nis known in mathematical biology as the fundamental theorem of natural selection [10, 17]\nand, in its original form, traces back to R. A. Fisher.\n\n+! \u0003\r\f;\u000b\n\u001fP\u0003\u0012\f;\u000b\n\u0003\u0012\f;\u000b\n\u0003\r\f;\u000b\b(\n\n\u0003\r\f;\u000b\n\n\u000b:\u0001\n\n\u0003\r\f;\u000b:\u0001\n\n!\u0016\u0015\n\n'\nG\n\n$\n7\n\n\n\u001f\n\u0001\n$\nG\n\u0004\n\u0007\n'\n\u0007\nG\nG\n\u0006\n\u001b\n&\nb\n\u001a\n\u001b\n&\n\u001c\n%\n\u0007\n&\nb\n&\n\u001a\n\u001b\n\u001f\n!\n\u000b\n\nG\n\nG\n\n\u001b\n$\n7\n\u001b\n$\n7\n\n\u001b\n\n\u001b\n$\n7\n\u000b\n\u0016\n\u000b\n\u000e\n7\n\u0014\n3\n\u0010\n\u0019\n\u001c\n%\n\u0010\n\u001a\n7\n3\n\n!\n\u0010\n!\n\u0010\n\u001f\n7\n\u0014\n3\nk\n\u001f\n!\n\n!\n$\n7\n\fis strictly increasing along any non-\nconstant trajectory under both continuous-time (4) and discrete-time (5) replicator dynam-\nics. Furthermore, any such trajectory converges to a stationary point. Finally, a vector\nis a strict local maximizer\n\nis asymptotically stable under (4) and (5) if and only if\n\nthen the function\n\n&\u000eb\n\nib\n\nTheorem 3 If\n\non\n\n.\n\nof\n\n&\u0018\u0010\n\n&4b\n\n$07\n\n\u0007\n\t\n\n\u0001W\u001a\n\nO0Q\n\n. By letting\n\n\u0001I\u0003\u0006\u0005.Q`\u0007P\tTQ-\u000b\n\nand\nmatrix of their FTAG\n\nIn light of their dynamical properties, replicator equations naturally suggest themselves\nas a simple heuristic for solving the maximal subtree isomorphism problem. Indeed, let\ndenote the adjacency\n\nbe two free trees, and let\n\nis the identity matrix, we know that the replicator dynamical systems (4) and (5),\nde\ufb01ned\nin (3) over the simplex and will eventually converge with probability 1 to a strict local\nmaximizer which, by virtue of Theorem 2, will then correspond to the characteristic vector\nof a maximal clique in the association graph. As stated in Theorem 1, this will in turn\ninduce a maximal subtree isomorphism between\n. Clearly, in theory there is no\n, and therefore that\n\n\u0001I\u0003h\u0005\nstarting from an arbitrary initial state, will iteratively maximize the function\u0006\nwhere\u0001\nguarantee that the converged solution will be a global maximizer of\u0006\n%\u0014\u0013\n\nRecently, there has been much interest around the following exponential version of repli-\ncator equations, which arises as a model of evolution guided by imitation [9, 10, 17]:\n\nit will induce a maximum isomorphism between the two original trees, but see below.\n\ntends to 0, the orbits of this dynamics approach those\n\nthe model approximates the so-called \u201cbest-reply\u201d dynamics [9, 10]. A\ncustomary way of discretizing equation (8) is given by the following difference equations:\n\nwhere\u0015\nof the standard, \u201c\ufb01rst-order\u201d replicator model (4), slowed down by the factor\u0015 ; moreover,\nfor large values of\u0015\n\n\u0003\r\f;\u000b\n\u0003\u0012\f;\u000b\b\u0001\nis a positive constant. As\u0015\n\u0003\r\f\n\nFrom a computational perspective, exponential replicator dynamics are particularly attrac-\ntive as they may be considerably faster and even more accurate than the standard, \ufb01rst-order\nmodel (see [13] and the experiments reported in the next section).\n\n\u0003\u0005\u0004\u0007\u0006\u0005\b\n\t\f\u000b\u000e\r\n\u0003\r\f;\u000b\n\u0003\r\f;\u000b\n\u0003\u0016\u0004\u0007\u0006\u0005\b\u0017\t\f\u000b\u000e\r\n\u0003\r\f;\u000b\n\u0004\u000f\u0006\n\n\u0004\u000f\u0006\u0007\u0010\u0011\t\u0012\u000b\u000e\n\n\t\u0012\u000b\u000e\n\n!\u0016\u0015\n\nand\n\nO0Q\n\n!\u0016\u0015\n\n\u000b:\u0001\n\n(7)\n\n(8)\n\n(9)\n\n4 Results and conclusions\n\nWe tested our algorithms over large random trees. Random structures represent a useful\nbenchmark not only because they are not constrained to any particular application, but also\nbecause it is simple to replicate experiments and hence to make comparisons with other\nalgorithms.\n\nIn this series of experiments, the following protocol was used. A hundred 100-node free\ntrees were generated uniformly at random using a procedure described by Wilf in [18].\nThen, each such tree was subject to a corruption process which consisted of randomly\ndeleting a fraction of its nodes (in fact, the to-be-deleted nodes were constrained to be\nthe terminal ones, otherwise the resulting graph would have been disconnected), thereby\nobtaining a tree isomorphic to a proper subtree of the original one. Various levels of corrup-\ntion (i.e., percentage of node deletion) were used, namely 2%, 10%, 20%, 30% and 40%.\nThis means that the order of the pruned trees ranged from 98 to 60. Overall, therefore, 500\n\n\u0001\n\n&\n$\n7\n&\n\n&\nO\n3\n3\n3\n\u000b\n\u001a\n\u001b\n\n\n\u001b\n\u001c\n%\n\u0007\n\u0001\n\u001b\nO\n3\n\u001b\n\u000b\n\n\u001f\n\n\u001f\n\u0002\n\u001a\n7\n3\n\n!\n\u0003\n\u0013\n\u0007\n\n\u001f\n\u001c\n%\n\n\u001f\n\u001a\n7\n3\n\n!\n\u0003\n\u0010\n(\n\fpairs of trees were obtained, for each of which the corresponding FTAG was constructed\nas described in Section 2. To keep the order of the association graph as low as possible, its\nvertex set was constructed as follows:\n\nassuming\u0004\n\n\u0005\u001d\u0001%\u0002r\u0003=\r\u000e\u0007\nkN\u000b\n\n\u0004 , the edge set\n\n\u0010@\u0005\n\n\u0017@\u0005\n\nbeing de\ufb01ned as in (2). It is straightforward to see\nthat when the \ufb01rst tree is isomorphic to a subtree of the second, Theorem 1 continues to\nhold. This simple heuristic may signi\ufb01cantly reduce the dimensionality of the search space.\nWe also performed some experiments with unpruned FTAG\u2019s but no signi\ufb01cant difference\nin performance was noticed apart, of course, heavier memory requirements.\n\nV\u0011)+*2,8\u0003/\r0\u000b\u0001\n\n*2,B\u0003/kN\u000b\n\nBoth the discrete-time \ufb01rst-order dynamics (5) and its exponential counterpart (9) (with\n\n' ) were used. The algorithms were started from the simplex barycenter and stopped\n\nwhen either a maximal clique (i.e., a local maximizer of\u0006\n\u0015\u0019\u0001\n\n) was found or the distance\nbetween two successive points was smaller than a \ufb01xed threshold. In the latter case the\nconverged vector was randomly perturbed, and the algorithms restarted from the perturbed\npoint. Note that this situation corresponds to convergence to a saddle point.\n\nAfter convergence, we calculated the proportion of matched nodes, i.e., the ratio between\nthe cardinality of the clique found and the order of the smaller subtree, and then we aver-\naged. Figure 1(a) shows the results obtained using the linear dynamics (5) as a function of\nthe corruption level. As can be seen, the algorithm was always able to \ufb01nd a correct maxi-\nmum isomorphism, i.e. a maximum clique in the FTAG. Figure 1(b) plots the correspond-\ning (average) CPU time taken by the processes, with corresponding error bars (simulations\nwere performed on a machine equipped with a 350MHz AMDK6-2 processor).\n\nIn Figure 2, the results pertaining to the exponential dynamics (8) are shown. In terms of\nsolution\u2019s quality the algorithm performed exactly as its linear counterpart, but this time it\nwas dramatically faster. This con\ufb01rms earlier results reported in [13].\n\nBefore concluding, we note that our approach can easily be extended to tackle the problem\nof matching attributed (free) trees. In this case, the attributes result in weights being placed\non the nodes of the association graph, and a conversion of the maximum clique problem to a\nmaximum weight clique problem [15, 5]. Moreover, it is straightforward to formulate error-\ntolerant versions of our framework along the lines suggested in [16] for rooted attributed\ntrees, where many-to-many node correspondences are allowed. All this will be the subject\nof future investigations.\n\nFinally, we think that the results presented in this paper (together with those reported in [13,\n15]) raise intriguing questions concerning the connections between (standard) notions of\ncomputational complexity and the \u201celusiveness\u201d of global optima in continuous settings.\n\nAcknowledgments. The author would like to thank M. Zuin for his support in performing\nthe experiments.\n\nReferences\n[1] D. H. Ballard and C. M. Brown. Computer Vision. Prentice-Hall, Englewood Cliffs, NJ, 1982.\n[2] H. Blum and R. N. Nagel. Shape description using weighted symmetric axis features. Pattern\n\nRecognition, 10:167\u2013180, 1978.\n\n[3] I. M. Bomze. Evolution towards the maximum clique. J. Glob. Optim., 10:143\u2013164, 1997.\n[4] I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In\nD.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization (Suppl. Vol. A),\npages 1\u201374. Kluwer, Boston, MA, 1999.\n\n[5] I. M. Bomze, M. Pelillo, and V. Stix. Approximating the maximum weight clique using repli-\n\ncator dynamics. IEEE Trans. Neural Networks, 11(6):1228\u20131241, 2000.\n\nb\nb\nb\n)\n\u0004\n\u0007\n\u0005\nb\n\u0004\n\n\u0004\n\u0005\nb\nb\n\t\n%\n\u001b\n\fFigure 1: Results obtained over 100-node random trees with various levels of corruption, using the\n\ufb01rst-order dynamics (5). Top: Percentage of correct matches. Bottom: Average computational time\ntaken by the replicator equations.\n\n[6] T. H. Cormen, C. E. Leiserson, and R. L. Rivest.\n\nCambridge, MA, 1990.\n\nIntroduction to Algorithms. MIT Press,\n\n[7] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-\n\nCompleteness. W. H. Freeman, San Francisco, CA, 1979.\n\n[8] S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching.IEEE Trans.\n\nPattern Anal. Machine Intell. 18:377-388, 1996.\n\n[9] J. Hofbauer. Imitation dynamics for games. Collegium Budapest, preprint, 1995.\n\n[10] J. Hofbauer and K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge\n\nUniversity Press, Cambridge, UK, 1998.\n\n[11] T.-L. Liu, D. Geiger, and R. V. Kohn. Representation and self-similarity of shapes. In Proc.\n\nICCV\u201998\u20146th Int. Conf. Computer Vision, pages 1129\u20131135, Bombay, India, 1998.\n\n[12] T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of Tur\u00b4an.\n\nCanad. J. Math., 17:533\u2013540, 1965.\n\n0\n0\n0\n3\n\n0\n0\n1\n\n0\n0\n5\n2\n\n0\n0\n0\n2\n\n0\n0\n5\n1\n\n5\n9\n\n0\n0\n0\n1\n\n0\n0\n5\n\nPercentage of correct matches\n\nAverage CPU time (in secs)\n\n\fFigure 2: Results obtained over 100-node random trees with various levels of corruption, using\nthe exponential dynamics (9) with\n\u0003\u0005\u0004 . Top: Percentage of correct matches. Bottom: Average\ncomputational time taken by the replicator equations.\n\n\u0002\u0001\n\n[13] M. Pelillo. Replicator equations, maximal cliques, and graph isomorphism. Neural Computa-\n\ntion, 11(8):2023\u20132045, 1999.\n\n[14] M. Pelillo and A. Jagota. Feasible and infeasible maxima in a quadratic program for maximum\n\nclique. J. Artif. Neural Networks, 2:411\u2013420, 1995.\n\n[15] M. Pelillo, K. Siddiqi, and S. W. Zucker. Matching hierarchical structures using association\n\ngraphs. IEEE Trans. Pattern Anal. Machince Intell., 21(11):1105\u20131120, 1999.\n\n[16] M. Pelillo, K. Siddiqi, and S. W. Zucker. Many-to-many matching of attributed trees using\nassociation graphs and game dynamics. In C. Arcelli, L. P. Cordella, and G. Sanniti di Baja,\neditors, Visual Form 2001, pages 583\u2013593. Springer, Berlin, 2001.\n\n[17] J. W. Weibull. Evolutionary Game Theory. MIT Press, Cambridge, MA, 1995.\n[18] H. Wilf. The uniform selection of free trees. J. Algorithms, 2:204\u2013207, 1981.\n[19] S. C. Zhu and A. L. Yuille. FORMS: A \ufb02exible object recognition and modeling system. Int. J.\n\nComputer Vision, 20(3):187\u2013212, 1996.\n\n0\n0\n4\n\n0\n0\n1\n\n0\n0\n3\n\n5\n9\n\n0\n0\n2\n\n0\n0\n1\n\nPercentage of correct matches\n\nAverage CPU time (in secs)\n\n\f", "award": [], "sourceid": 2057, "authors": [{"given_name": "Marcello", "family_name": "Pelillo", "institution": null}]}