A. Jagota. Efficiently approximating MAX-CLIQUE in a hopfield-style network. In Proc. of Int. Joint Conf. on Neural Networks '92 Vol. II, pages 248--253, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 iff ae(n x i ; n y k ) and o ik = 0 otherwise. So, the objective function of the optimization tasks is f(o ) max o2f0;1g jNx j Thetaf0;1g jNy j Nx X i;j=1 Ny X k;l=1 w ij;kl o ik o jl (9) where w ij;kl = 8 : w Gammaw I 0 9 = according to Eq. 6) It has been shown ([18, 17, 2, 19, 24, 5, 12, 36]) that such quadratic optimization tasks can be solved by Hopfield like Artificial Neural Nets, i.e. bi directional associative memories. In contrast to the most other implementations which use Hopfield Nets with binary or continuous sigmoidal output function, in this work an approach described ....

A. Jagota. Efficiently approximating MAX-CLIQUE in a hopfield-style network. In Proc. of Int. Joint Conf. on Neural Networks '92 Vol. II, pages 248--253, 1992.

....1 iff ae(n x i ; n y k ) and o ik = 0 otherwise. So, the objective function of the optimization tasks is f(o ) max o2f0;1g jNx j Thetaf0;1g jNy j Nx X i;j=1 Ny X k;l=1 w ij;kl o ik o jl (7) where w ij;kl = 8 : w Gammaw I 0 9 = according to Eq. 4) It has been shown ([20, 19, 2, 21, 28, 7, 14, 39]) that such quadratic optimization tasks can be solved by Hopfield like Artificial Neural Nets, i.e. bi directional associative memories. In contrast to the most other implementations which use Hopfield Nets with binary or continuous sigmoidal output function, in this work an approach described ....

A. Jagota. Efficiently approximating MAX-CLIQUE in a hopfield-style network. In Proceedings of International Joint Conference on Neural Networks '92 Volume II, pages 248--253, 1992.

.... Fast methods of clique determination exist for smaller graphs and graphs of low density, for example [BP95, dABR92, Bab94, Bab91] For a comprehensive overview see [PX93] Clique determination by means of BAMs like the one presented in this paper are published, for instance, in [FTL92, Jag90] and [Jag92]. But before the CG can be mapped on a Neural Net, the definition of the CG must be extended. The existing edges which connect compatible match hypotheses are weighted by some positive number w. In addition, another set of edges is introduced. The new edges connect the incompatible hypotheses. ....

Arun Jagota. Efficiently approximating MAX-CLIQUE in a hopfield-style network. In Proceedings of International Joint Conference on Neural Networks '92 Volume II, pages 248--253, 1992. ftp dimacs.rutgers.edu pub/challenge/graph/contributed.

....and the theory of neural networks. Starting from the early work of Hopfield [42] and Hopfield and Tank [43] several approaches have been developed for a variety of applications. Akiyama et al. 44] introduced Gaussian machines for optimization. Jagota concentrated on the maximum clique problem [45, 46], which was considered by Funabiki et al. 47] and others as well. Gold, Rangarajan et al. 48, 49, 50, 51, 52] developed sophisticated, fast methods for graph matching by neural networks and by the softassign method. Suganthan et al. 53, 54, 41] introduced a neural net approach for finding one ....

Arun Jagota. Efficiently approximating MAXCLIQUE in a hopfield-style network. In Proceedings of International Joint Conference on Neural Networks '92 Volume II, pages 248--253, 1992.

....with the required cost with at least probability (1 Gamma ffl) unless R = NP [Joh92, p. 519] The results in this section apply only to the restricted class of neural networks considered in [BG90] Less constrained types of neural networks may exist for these problems; for example, Jagota [Jag92] has designed asymmetric weighted neural networks for MAX CLIQUE that perform extremely well on average. 5.6 Summary The known theoretical and algorithmic lower limits on approximability for the phylogenetic inference problems examined in this thesis are given in Table 20. Though the ....

Jagota, A. Efficiently Approximating MAX-CLIQUE in a Hopfield-style Network. To appear in International Joint Conference on Neural Networks, IEEE Computer Society Press, Washington, D. .C, 1992.

....graphs stemming from numerous applications. Indeed, many different types of algorithmic approaches are applied to that problem. Several neural networks and related algortihms were applied recently to combinatorial optimization problems in general (see e.g. 8, 6, 19] and to the Max Clique problem [1, 2, 5, 10, 11, 15, 18] in particular. These neural nets are dynamical system which minimize a cost (or computational energy ) function that represents the optimization problem, the MaxClique in our case. Therefore they all belong to the class of integer programming algorithms surveyed in the Pardalos and Xue review ....

.... the Pardalos and Xue review [13] The work presented here is a development and improvement of a neural network algorithm that was introduced recently [5] In the previous work, we have considered two Hopfield type neural networks, the INN and the HcN (which was applied before in that context, see [10, 11]) and their application to the max clique problem. In this paper, I concentrate on the INN network and present an improved version of the t Annealing (t A) algorithm that was introduced in [5] The rest of this paper is organaized as follows: in section 2 I describe the INN model and how it ....

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A. Jagota. Efficiently approximating Max-Clique in a Hopfield-style network. In International Joint Conference on Neural Networks, volume 2, pages 248--253, New York, June 1992. Baltimore, June, IEEE.

....with the aid of HcN. 3. 2 HcN as a way to solve the Max Clique problem In 1990 Jagota introduced a variant of HN, called Hopfield style Network (after 1992 referred to as Hopfield clique Network ) built in such a way that its stable states are exactly the maximal cliques of the underlying graph ([Jag92]) Definition. A discrete asynchronous Hopfield Network of N units ( Y i 2 f0; 1g ) with weights W ij 2 f GammaK; 1g, K 0) W ii = 0, and bias B i 2 (0; 1) is called Hopfield clique Network (HcN) HcN is characterized by its underlying graph, i.e. by graph GN = V; E) whose vertices are the ....

....will not allow the state depicted on Fig.3 (b) to be stable. So we have Maximality Condition. If :9 unit i: switches OFF ON and YON is a clique, then YON is a maximal clique. The small bias b i prevents the values at the right side of equations above to be equal to zero. fl In [Gros92] and in [Jag92] the new type of HcN dynamics called Steepest Ascent Dynamics was proposed. It is very similar to the Steepest Descent dynamics, except for that the unit i is first picked and then switched, if its switch would maximally increase the energy. We have another theorem for such a dynamics( Jag.2] ....

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A. Jagota. Efficiently approximating Max-Clique in a Hopfield-style network. International Joint Conference on Neural Networks, volume 2, pages 248--253, New York, June 1992. Baltimore, June, IEEE.

....cliques on average than exact algorithms or more sophisticated heuristics but run considerably faster. All our algorithms are simple and inherently parallel. 1 Introduction In earlier work, we applied several neural network heuristics to the approximate solution of the maximum clique problem [Jag92, JR92, Jag93] In the current paper, we present improvements to our previous work as well as extensions to new related algorithms and their experimental evaluation. The focus of our improvements is the following. In our previous studies, the two heuristics that performed the best, in terms of ....

....weight of the edge fv i ; v j g. Vertices have binary valued (in f0; 1g) or continuous (in [0; 1] labels S i on them. I = I i ) is the vector of external biases to vertices. We refer the reader to [HKP91] for further details. We encode the Maximum Clique problem in a Hopfield Network as follows [Jag92] Given an N vertex graph G, construct a Hopfield network instance H with N vertices, one for each vertex v i in G. Let w ij = 1 if vertices v i ; v j are adjacent in G, w ij = ae otherwise. We normally choose ae = Gamma4N . Let I = w 0 ) where w 0 = jaej=4. Earlier encodings of the Maximum ....

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A. Jagota. Efficiently approximating Max-Clique in a Hopfield-style network. In International Joint Conference on Neural Networks, volume 2, pages 248--253, New York, June 1992. Baltimore, June, IEEE.

....results are reported. Their encoding admits invalid solutions and employs non binary weights. The Maximum Independent Set problem is indirectly formulated in [37] with essentially the same encoding as in [11] No experiments are reported. The approach of the current paper was presented in [20] and further studied in [23] The weights are binary and no invalid solutions are admitted. Recently, another encoding of Maximum Clique is in [30] The weights are non binary. Good experimental performance is reported on random graphs. Our Mean Field Annealing and stochastic dynamics performances ....

.... DeltaE i = min j DeltaE j 0 where DeltaE k (t) Gamma[S k (t) Gamma S k (t Gamma 1) n k is the energy change caused by the switch (i.e. S k (t) 6= S k (t Gamma 1) of S k . With ae Gamma2N , SD emulates the following greedy clique finding algorithm on the underlying graph GN [20]: 1. S S 0 ; 2. while S is not a clique of GN do (a) Pick a v 2 S with minimum degree in G[S] i.e. d G[S] v = ffi (G[S] b) S S n fvg; 3. end; 4. while S is not a maximal clique of GN do (a) Pick v 2 V n S such that S [ fvg is a clique of GN ; b) S S [ fvg; 5. end; SD(S 0 ; ae ....

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A. Jagota. Efficiently approximating max-clique in a Hopfield-style network. In International Joint Conference on Neural Networks, volume 2, pages 248--253, New York, June 1992. Baltimore, June, IEEE.

.... given graph and is NP hard [22] even to approximate [4] In a recent paper, we have approximately solved some NP hard problems by reduction to MAX CLIQUE followed by approximate solution of MAX CLIQUE in a Hopfield network special case (HcN) whose stable states are maximal cliques [20] also see [18]) In the current work, we add a new problem, solved in a similar way, to this list: graph k coloring. The reduction of graph k coloring to MAX CLIQUE described here has the following property: every maximal clique in a reduced instance corresponds to some feasible solution of the original ....

....rules, updates according to any of which monotonically decrease the energy function E = Gamma 1 2 S T WS Gamma I T S [13] thus guaranteeing eventual convergence to a discrete local minimum of it. Here I = I i ) is the vector of external biases to units. The Hopfield clique Network (HcN) [17, 18] is a binary weights special case of the Hopfield Network. We describe the n unit HcN in equivalent form [12] that has been implemented in optics [26] W = f Gamma1; 0g n Thetan is the symmetric and zero diagonal weight matrix. I = b n is the vector of external unit biases; 0 b 1 2 . GN = ....

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A. Jagota. Efficiently approximating max-clique in a Hopfield-style network. In International Joint Conference on Neural Networks, volume 2, pages 248--253, New York, June 1992. Baltimore, June, IEEE.

....raised above. ffl The good performance under q of deterministic greedy (i.e. steepest) descent from the full vertex set V to a clique, called SD(V ) seems surprising to us. SD(V ) did nearly as well as mean field annealing (MFA) BMM 89] and a variant called ae annealing introduced in [Jag92] and better than continuous Hopfield dynamics (CHD) Hop84] As expected, it outperformed one run of stochastic steepest descent (SSD(V; 1) but it did only very slightly worse than N : jV j runs of the latter (SSD(V; N) For details on the heuristics and their performance, see Section 5. 2. ....

....= 0:5, d p (N ) is quite close to c p (N ) even for small N , but for p = 0:9 and the values of N we consider it is a serious underestimate: d :9 (100) 22:53 not 34.86, d :9 (400) 43:86 not 55.22. 5. The Neural Network Heuristics Hopfield clique Network. The Hopfield clique Network (HcN) Jag92] is a binary weights special case of the symmetric weights zero diagonal Hopfield Network (HN) Hop82, Hop84] HcN imposes the following restriction on the weights of HN: for i 6= j, w ij 2 fae; 1g where ae 0. In the discrete version, unit states are: S i 2 f0; 1g. For ae GammaN and 0 w 0 ....

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A. Jagota. Efficiently approximating max-clique in a hopfield-style networ k. In International Joint Conference on Neural Networks, volume 1, New York, 1992. Baltimore, June, IEEE.

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