I. M. Bomze, M. Pelillo, and V. Stix, Approximating the maximum weight clique using replicator dynamics. IEEE Trans. on Neural Networks, 11(6):1228--1241, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a special case of the preceding result. As in the unweighted case, this formulation suffers from the existence of spurious solutions, and this entails the lack of a one to one cor respondence between the solutions of the continuous optimization problem and those of the original, discrete one. In [11] these spurious so lutions are characterized and a regularized version which avoids this kind of problems is in troduced (see also [7] Specifically, let Af(G, w) be the the class of n x n symmetric matrices M = mij)i,jv defined as mij mii q mjj if 1 (i, j) E and mij = 0 otherwise, and mii ....

....where 7 is an arbitrary constant. Therefore, the payoff matrix for replicator dynamics to be used in this case is: W = Tee T M where M = mij) is any matrix in JV (G, w) and 7 = max mij . i,jv Experiments with this approach on both random graphs and DIMACS benchmark graphs are reported in [11]. Weights were generated randomly in both cases. The results obtained with replicatot dynamics (3) were compared with those produced by a very efficient maximum weight clique algorithm of the branchand bound variety. The algorithm performed remarkably well especially on large and dense graphs, ....

BOMZE, I. M., PELILLO, M., AND STIX, V.: Approximating the maximum weight clique using replicatot dynamics, Tech. Rep. CS-99-13, Dipartimento di Informatica, Universirk Ca' Foscari di Venezia, 1999.

....at problem s order density 1000 10 respectively 363.60 seconds at problem s order density 1000 30 . In case that problem s parameters exceed those constraints (especially rescaling weights as real numbers can lead to unhandy large colors) we might move to a method recently announced in [BPS00] which is centered around a continous formulation of MWCP . In the sixtees it has been shown that the clique number of a graph is connected with optimizing a certain quadratic program. Iterating so called Replicator Equations over the standard simplex [BPS00] proves the reach at a (local) ....

....to a method recently announced in [BPS00] which is centered around a continous formulation of MWCP . In the sixtees it has been shown that the clique number of a graph is connected with optimizing a certain quadratic program. Iterating so called Replicator Equations over the standard simplex [BPS00] proves the reach at a (local) solution, according their experiments at least 10 times faster than the above algorithm. Since this method produces definitely local solutions (i.e. maximal weight cliques) it qualifies itself for denser problems where local and optimal solutions grow rich together. ....

BOMZE, I. M. ; PELILLO, M. ; STIX, V.: Approximating the Maximum Weight Clique Using Replicator Dynamics. In: IEEE Transactions on Neural Networks 11 (2000), November, Nr. 6

....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] In this paper, we first introduce a wide family of payoffmonotonic dynamical systems of which replicator equations are just a special instance. The models in this family enjoy precisely the same dynamical properties as replicator equations, and hence they naturally suggest ....

I. M. Bomze, M. Pelillo, and V. Stix, "Approximating the maximum weight clique using replicator dynamics," IEEE Trans. Neural Networks, vol. 11, no. 6, pp. 1228--1241, 2000.

....we note that our approach can easily be extended to tackle the problem of matching attributed (free) trees. In this case, the attributes result in weights being placed on the nodes of the association graph, and a conversion of the maximum clique problem to a maximum weight clique problem [15, 5]. Moreover, it is straightforward to formulate errortolerant versions of our framework along the lines suggested in [16] for rooted attributed trees, where many to many node correspondences are allowed. All this will be the subject of future investigations. Finally, we think that the results ....

I. M. Bomze, M. Pelillo, and V. Stix. Approximating the maximum weight clique using replicator dynamics. IEEE Trans. Neural Networks, 11(6):1228--1241, 2000.

.... The following theorem expands on a recent result by Gibbons et al. 8] which in turn generalizes the so called Motzkin Straus theorem [15] This allows us to formulate the maximum weight clique problem as a quadratic program, thereby switching from the discrete to the continuous domain (see [5] for proof) Theorem 2 Let G = V; E; be an arbitrary weighted graph and consider a matrix A 2 M(G) Then, the following hold: a) A vector x 2 Sn is a local minimizer of f on Sn if and only if x = x c , where C is a maximal weight clique of G. b) A vector x 2 Sn is a global minimizer of ....

....G. b) A vector x 2 Sn is a global minimizer of f on Sn if and only if x = x c , where C is a maximum weight clique of G. Moreover, all local (and hence global) minimizers of f on Sn are strict. Unlike the Gibbons et al. s formulation, which is plagued by the presence of spurious solutions [5], the previous result guarantees that all minimizers of f on Sn are strict, and are characteristic vectors of maximal maximum cliques in the graph. In a formal sense, therefore, a oneto one correspondence exists between maximal maximum weight cliques and local global minimizers of f on Sn A ....

I. M. Bomze, M. Pelillo, and V. Stix. Approximating the maximum weight clique using replicator dynamics. Technical Report CS-99-13, University of Venice, Dept. of Computer Science, June 1999.

....other words, the Motzkin Straus theorem turns out to be a special case of the preceding result. As in the unweighted case, the existence of spurious solutions entails the lack of one toone correspondence between the solutions of the continuous problem and those of the original, discrete one. In [16] these spurious solutions are characterized and a regularized version which avoids this kind of problems is introduced, exactly as in the unweighted case (see also [12] Replicator equations are then used to find maximal weight cliques in weighted graphs, using this formulation. Experiments with ....

Bomze, I. M., Pelillo, M., and Stix, V.: Approximating the maximum weight clique using replicator dynamics, Tech. Rep. CS-99-13, Dipartimento di Informatica, Universit`a Ca' Foscari di Venezia, 1999.

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I. M. Bomze, M. Pelillo, and V. Stix, Approximating the maximum weight clique using replicator dynamics. IEEE Trans. on Neural Networks, 11(6):1228--1241, 2000.

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