**10. Acknowledgement**

I would like to acknowledge my profound gratitude to the Management, Rashtreeya Sikshana Samithi Trust, Bangalore.I am indebted to Dr. S.C. Sharma, Vice Chancellor, Tumkur University, Karnataka for his unending support.I would like to thank all my official colleagues at R V College of Engineering and specifically the MCA department staffs for their remarkable co-ordination. Last but not the least, I would like to thank my family members.

Algorithms for CAD Tools VLSI Design 165

Carpenter, G.A., Grossberg, S., "Adaptive Resonance Theory, In Michael A. Arbib" (Ed.),

Grossberg, S., "Competitive learning: From interactive activation to adaptive resonance",

Carpenter, G.A, Grossberg, S., "ART 2: Self-organization of stable category recognition codes for analog input patterns", *Applied Optics, 26(23)*, 1987, pp. 4919-4930. Carpenter, G.A., Grossberg, S., Reynolds J.H., "ARTMAP: Supervised real-time learning and

Carpenter, G.A., Grossberg, S., Rosen, D.B., "Fuzzy ART: Fast stable learning and

Carpenter, G.A., Grossberg, S., Markuzon, N., Reynolds, J.H., Rosen D.B., "Fuzzy ARTMAP:

A. Canuto, G. Howells, M. Fairhurst, "An investigation of the effects of variable vigilance

A. Dubrawski, "Stochastic validation for automated tuning of neural network's hyper-

P. Gamba and F. DellAcqua, "Increased accuracy multiband urban classification using a

 C. P. Lim, H. H. Toh, T. S. Lee, "An evaluation of the fuzzy ARTMAP neural network using offline and on-line strategies," *Neural Network World, 4*, 1999, pp. 327-339. Carpenter, G.A., Grossberg, S., Markuzon, N., Reynolds, J.H., and Rosen, D.B., "Fuzzy

Caudell, T.P., Smith, S.D.G., Escobedo, R., Anderson, M, "NIRS: Large scale ART–1 neural

Benno Stein, Michael Busch, "Density based cluster Algorithms in Low-dimensional and

Busch. Analyse dichtebasierter Clusteralgorithmen am Beispiel von DBSCAN und

Martin Ester, Hans-Peter Kriegel, Jörg Sander, Xiaowei Xu, "A density-based algorithm for

parameters," *Robotics and Automated Systems*, 21, 1997, pp. 83-93.

MIT Press, 2003, pp. 87 – 90.

*Networks (Publication),* 4, 1991, pp. 565-588.

*Networks (Publication),* 4, 1991, pp. 759-771.

Systems, 29:4, 2000, pp. 317-334.

834.

pp. 698-713.

March 2005.

Press, 1996, pp. 226-231.

1350.

Cognitive Science

*The Handbook of Brain Theory and Neural Networks*, Second Edition,Cambridge, MA:

classification of nonstationary data by a self-organizing neural network", *Neural* 

categorization of analog patterns by an adaptive resonance system", *Neural* 

A neural network architecture for incremental supervised learning of analog multidimensional maps", *IEEE Transactions on Neural Networks*, 3, 1992, pp. 698-713.

within the RePART neuro-fuzzy network," Journal of Intelligent and Robotic

neuro-fuzzy classifier," *International Journal of Remote Sensing*, 24:4, 2003, pp. 827-

ARTMAP: neural network architecture for Incremental supervised learning of analog multidimensional maps", *IEEE Transactions on Neural Networks*, 1992, 3:,

architectures for engineering design Retrieval", *Neural Networks, 7:*, 1994, pp. 1339-

High-dimensional Applications", *Second International Workshop on Text- Based information Retrieval (TIR 05)*, Stein, Meyer zu Eißen (Eds.) Fachbericht,e, Informatik, University of Koblenz-Landau, Germany, ISSN 1860-4471c, pp. 45-56.

MajorClust. Study work, Paderborn University, Institute for Computer Science,

discovering clusters in large spatial databases with noise". *Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD-96)*: AAAI

#### **11. References**


J. B. MacQueen, "Some Methods for classification and Analysis of Multivariate

Shantanu Dutt , Wenyong Deng, "A probability-based approach to VLSI circuit

N. Krasnogor, J. Smith, "Memetic algorithms: The polynomial local search complexity

J. T., "Evolutionary Algorithms, Fitness Landscapes and Search". *Ph.D. thesis, Univesity of* 

P. Merz and B. Freisleben, "Fitness landscapes, memetic algorithms, and greedy operators

T. Jones, "Crossover, macromutation, and population based search," *in Proceedings of the* 

N. Krasnogor, P. M. L´opez, E. de la Canal, D. Pelta, "Simple models of protein folding and

Christopher M. Bishop, "Neural Networks for Pattern Recognition", *Oxford Univ Press*,

N. Krasnogor, P. Mocciola, D. Pelta, G. Ruiz, W. Russo, "Arunnable functional memetic

D. Holstein, P. Moscato, "Memetic algorithms using guided local search: A case study," *in* 

Dimitris Karlis, "An EM algorithm for multivariate Poisson distribution and related models", *Journal of Applied Statistics*, Vol. 30, No. 1, 2003 , pp. 63-77(15) Carpenter, G.A, "Distributed learning, recognition, and prediction by ART and ARTMAP

Sung Ho Kim, "Calibrated initials for an EM applied to recursive models of categorical

Gyllenberg M, Koski T, Lund T, "Applying the EM-algorithm to classification of bacteria",

variables", *Computational Statistics & Data Analysis*, Vol. 40, Issue 1, July 2002, pp.

*Proceedings of the International ICSC Congress on Intelligent Systems and Applications*,

*Operations Research: Recent Advances in the Interface meeting*, 1998.

*Sciences*. Universidad Nacionaldel Comahue, 1998, pp. 525–536.

neural networks", *Neural Networks*, 10:, 1997, pp. 1473- 494.

*Edition, Morgan Kaufmann*, San Francisco, 2005.

*Probability*, Berkeley, University of California Press, 1967, pp. 281-297. Ata Kaban, Mark Girolami, "Initialized and Guided EM-Clustering of Sparse Binary Data

*Pattern Recognition (ICPR'00)*, Vol. 2, 2000, pp. 2744.

*New Mexico*, Albuquerque, NM, 1995.

Vegas, Nevada, United States, June 03-07, 1996 , pp.100-105.

Observations", *Proceedings of 5-th Berkeley Symposium on Mathematical Statistics and* 

with Application to Text Based Documents," ICPR, *15th International Conference on* 

partitioning", *Proceedings of the 33rd annual conference on Design automation*, Las

theory perspective," *Journal of Mathematical Modelling and Algorithms*, Springer Netherlands, ISSN: 1570-1166 (Print) 1572-9214 (online), Vol. 7, Number 1 / March,

for graph bipartitioning," *Journal of Evolutionary Computation*, vol. 8, no. 1, 2000, pp.

*Sixth International Conference on Genetic Algotihms*, M. Kauffman, Ed., 1995, pp. 73–

a memetic crossover," *in Exposed at INFORMS CSTS, Computer Science and* 

algorithm framework," in *Proceedings of the Argentinian Congress on Computer* 

*New Ideas in Optimization*, D. Corne, F. Glover, M. Dorigo, Eds. McGraw-Hill, 1999. Arthur Dempster, N Laird, D B Rubin, "Maximum likelihood estimation for incomplete data via the EM algorithm", Journal of the *Royal Statistical Society* (B) 39, 1977, pp.1-38. Witten I, Frank E, "Data Mining: Practical Machine Learning Tools and Techniques", *Second* 

**11. References** 

2008, pp. 3 - 24.

ISBN: 0198538642,1995.

61–91.

80.

97-110 .

2000, pp. 65-71.


**1. Introduction**

form:

**1.1 The satisfiability problem**

propositional formula Φ = �*<sup>m</sup>*

guidelines for future work.

The satisfiability problem (SAT) which is known to be NP-complete (7) plays a central role problem in many applications in the fields of VLSI Computer-Aided design, Computing Theory, and Artificial Intelligence. Generally, a SAT problem is defined as follows. A

Boolean variable, *xi*, *i* ∈ {1, . . . , *n*}, takes one of the two values, *True* or *False*. A clause , in turn, is a disjunction of literals and a literal is a variable or its negation. Each clause *Cj* has the

where *Ij*, ¯*Ij* ⊆ {1, .....*n*}, *<sup>I</sup>* <sup>∩</sup> ¯*Ij* <sup>=</sup> <sup>∅</sup>, and *<sup>x</sup>*¯*<sup>i</sup>* denotes the negation of *xi*. The task is to determine whether there exists an assignment of values to the variables under which Φ evaluates to *True*. Such an assignment, if it exists, is called a satisfying assignment for Φ, and Φ is called satisfiable. Otherwise, Φ is said to be unsatisfiable. Since we have two choices for each of the *<sup>n</sup>* Boolean variables, the size of the search space *<sup>S</sup>* becomes <sup>|</sup>*S*<sup>|</sup> <sup>=</sup> <sup>2</sup>*n*. That is, the size of the search space grows exponentially with the number of variables. Since most known combinatorial optimization problems can be reduced to SAT (8), the design of special methods for SAT can lead to general approaches for solving combinatorial optimization problems. Most SAT solvers use a Conjunctive Normal Form (CNF) representation of the formula Φ. In CNF, the formula is represented as a conjunction of clauses, with each clause being a disjunction of literals. For example, *P* ∨ *Q* is a clause containing the two literals *P* and *Q*. The clause *P* ∨ *Q* is satisfied if either *P* is *True* or *Q* is *True*. When each clause in Φ contains

The rest of the paper is organized as follows. Section 2 provides an overview of algorithms used for solving the satisfiability problem. Section 3 reviews some of the multilevel techniques that have been applied to other combinatorial optimization problems. Section 4 gives a general description of memetic algorithms. Section 5 introduces the multilevel memetic algorithm. Section 6 presents the results obtained from testing the multilevel memetic algorithm on large industrial instances. Finally, in Section 7 we present a summary and some

⎛ <sup>⎝</sup> � *<sup>l</sup>*∈¯*Ij x*¯*l* ⎞ ⎠ ,

*Cj* =

exactly *k* literals, the resulting SAT problem is called *k*-SAT.

⎛ <sup>⎝</sup> � *k*∈*Ij xk* ⎞ ⎠ ∨

*<sup>j</sup>*=<sup>1</sup> *Cj* with *m* clauses and *n* Boolean variables is given. Each

**A Multilevel Approach Applied to** 

Noureddine Bouhmala *Vestfold University College*

**Sat-Encoded Problems** 

*Norway*

**8**

