Abstract
The need to solve optimization problems of unprecedented sizes is becoming a challenging task. Utilizing classical methods of Operations Research often fail due to the exponentially growing computational effort. It is commonly accepted that these methods might be heavily penalized by the NP-Hard nature of the problems and consequently will then be unable to solve large size instances of a problem. Lacking the theoretical basis and guided by intuition, meta-heuristics are the techniques commonly used even if they are unable to guarantee an optimal solution. Meta-heuristics search techniques tend to spend most of the time exploring a restricted area of the search space preventing the search to visit more promising areas thereby leading to solutions of poor quality. In this paper, a multilevel learning automata and a multilevel WalkSAT algorithm are proposed as a paradigm for finding a tactical interplay between diversification and intensification for large scale optimization problems. The multilevel paradigm involves recursive coarsening to create a hierarchy of increasingly smaller and coarser versions of the original problem. This phase is repeated until the size of the smallest problem falls below a specified reduction threshold. A solution for the problem at the coarsest level is generated, and then successively projected back onto each of the intermediate levels in reverse order. The solution at each child level is improved before moving to the parent level. Benchmark including large MAX-SAT test cases are used to compare the effectiveness of the multilevel approach against its single counter part.
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References
Benlik U, Hao J-K (2011) A multilevel memetic approach for improving graph k-partitions. Evol Comput IEEE Trans 15(5):624–642
Blum C, Puchinger J, Raidl GR, Roli A (2011) Hybrid metaheuristics in combinatorial optimization: a survey. Appl Soft Comput 11:4135–4151
Biere A, Cimatti A, Clarke E, Zhu Y (1999) Symbolic model cheking without BDDs. In: Tools and algorithms for the construction and analysis of systems, pp 193–207
Boughaci D, Benhamou B, Drias H (2008) Scatter search and genetic algorithms for MAX-SAT problems. J Math Model Algorithm, pp 101–124
Boughaci D, Drias H (2005) Efficient and experimental meta-heuristics for MAX- SAT problems. In: Lecture notes in computer sciences, WEA 2005,3503/2005, pp 501–512
Bouhmala N (2012) A multilevel memetic algorithm for large sat-encoded problems. Evolutionary computation, MIT Press Cambridge, USA 20(4):641–664
Bouhmala N, Granmo OC (2011) GSAT enhanced with learning automata and multilevel paradigm. Int J Comput Sci 8(3)
Bouhmala N, Salih S (2012) A multilevel tabu search for the maximum satisfiability problem. Int J Commun Netw Syst Sci 5:661–670
Cai S, Luo C, Su K (2012) CCASat: solver description. In: Proceedings of SAT challenge 2012: solver and benchmark descriptions. pp 13–14
Cai S, Su K, Sattar A (2011) Local search with edge weighting and configuration checking heuristics for minimum vertex cover. Artif Intell 175(9–10):1672–1696
Cha B, Iwama K (1995) Performance tests of local search algorithms using new types of random CNF formula. In: Proceedings of IJCAI95. Morgan Kaufmann Publishers, pp 304–309
Cook SA (1971) The complexity of theorem-proving procedures. In: Proceedings of the third ACM symposium on theory of computing, pp 151–158
Drias H, Douib A, Hireche C (2013) Swarm intelligence with clustering for solving SAT. Lect Notes Comput Sci 8206:585–593
Frank J (1997) Learning short-term clause weights for GSAT. In: Proceedings of IJCAI97, Morgan Kaufmann Publishers, pp 384–389
Glover F, Kochenberger GA (2003) Handbook of metaheuristics, Springer
Granmo OC, Bouhmala N (2007) Solving the satisfiability problem using finite learning automata. Int J Comput Sci Appl 4(3):15–29
Hadany R, Harel D (1999) A multi-scale algorithm for drawing graphs nicely. Tech Rep CS99-01, Weizmann Inst Sci, Faculty Maths Comp Sci
Hansen P, Jaumard B, Mladenovic N, Parreira AD (2000) Variable neighborhood search for maximum weighted satisfiability problem. Technical Report G-2000-62, Les Cahiers du GERAD, Group for Research in Decision Analysis
Hendrickson B, Leland R (1995) A multilevel algorithm for partitioning graphs. In: Karin S, (ed), Proceedings Supercomputing’95, San Diego, ACM Press, New York
Holland JH (1975) Adaptation in natural and srtificial systems. University of Michigan Press, Ann Arbor
Hoos H (2002) An adaptive noise mechanism for WalkSAT, In: Proceedings of AAAI-2002, pp 655–660
Hoos H, Stützle T (2000) Local search algorithms for SAT. An empirical evaluation. J Automat Reason 24:421–481
Hoos H (1999) On the run-time behavior of stochastic local search algorithms for SAT. In: Proceedings of AAAI-99, pp 661–666
Jin-Kao H, Lardeux F, Saubion F (2003) Evolutionary computing for the satisfia- bility problem. In: Applications of evolutionary computing, volume 2611 of LNCS, University of Essex, England, pp 258–267
KhudaBukhsh AR, Xu L, Hoos HH, Leyton-Brown K (2009) SATenstein: automatically building local search SAT solvers from components. In: Proceedings of the 25th international joint conference on artificial intelligence (IJCAI-09)
Laguna M, Glover F (1999) Scatter search. Graduate school of business, University of Colorado, Boulder
Lardeux F, Saubion F, Jin-Kao H (2006) GASAT: a genetic local search algorithm for the satisfiability problem. Evol Comput, MIT Press 14(2)
Li CM, Wei W, Zhang H (2007) Combining adaptive noise and look-ahead in local search for SAT. Lect Notes Comput Sci 4501:121–133
Li CM, Huang WQ (2005) Diversification and determinism in local search for satisfiability. In: Proceedings of the eighth international conference on theory and applications of satisfiability testing (SAT-05), volume 3569 of lecture notes in computer science, pp 158–172
Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24:1097–1100
Karypis G, Kumar V (1998) Multilevel k-way partitioning scheme for irregular graphs. J Par Dist Comput 48(1):96–129
Karaboga D, Akay B (2009) A comparative study of artificial bee colony algorithm. Appl Math Comput 214:108–132
Mazure B, \(Sa\ddot{i}s\) L, \(Gr\acute{e}goire\) E (1997) Tabu search for SAT. In: Proceedings of the fourteenth national conference on artificial intelligence (AAAI-97), pp 281–285
McAllester D, Selman B, Kautz H (1997) Evidence for invariants in local search. In: Proceedings of the fourteenth national conference on artificial intelligence (AAAI-97), pp 321–326
Narendra KS, Thathachar MAL (1989) Learning automata: an introduction. Prentice Hall
Oduntan IO, Toulouse M, Baumgartner R, Bowman C, Somorjai R, Crainic TG (2008) A multilevel tabu search algorithm for the feature selection problem in biomedical data. Comput Math Appl 55(5):1019–1033
Rintanen J, Heljanko K, Niemelä I (2006) Planning as satisfiability: paralel plans and algorithms for plan search. Artif Intell, 170(12–13):1031–1080
Selman B, Kautz HA, Cohen B (1994) Noise strategies for improving local search. In: Proceedings of AAAI’94, MIT Press, pp 337–343
Selman B, Kautz H, Cohen B (1994) Noise strategies for improving local search. In: Proceedings of national Conference on artificial intelligence (AAAI)
Selman B, Levesque H, Mitchell D (1992) A new method for solving hard satisfiability problems. In: Proceedings of AAA92, MIT Press, pp 440–446
Smith A, Veneris AG, Ali MF, Viglas A (2005) Fault diagnosis and logic debugging using Boolean satisfiability. IEEE Trans Comput Aided Des 24(10):1606–1621
Smyth K, Hoos H, Stutzle T (2003) Iterated robust tabu search for MAX-SAT. Lect Notes Artif Intell 2671:129–144
Thathachar MAL, Sastry PS (2004) Network of learning automata: techniques for Online stochastic optimization. Kluer Academic Publishers
Tsetlin ML (1973) Automaton theory and modeling of biological systems. Academic Press
Yagiura M, Ibaraki T (2001) Efficient 2 and 3-flip neighborhood search algorithms for the MAX SAT: experimental evaluation. J Heuristics 7:423–442
Walshaw C (2003) A multilevel algorithm for forced-directed graph-drawing. J Graph Algorithm Appl 7(3):253–285
Walshaw C (2002) A multilevel approach to the traveling salesman problem. Oper Res 50(5):862–877
Walshaw C (2001) A multilevel Lin–Kernighan–Helsgaun algorithm for the travel-ling salesman problem. Tech Rep 01/IM/80, Comp Math Sci, Univ. Greenwich
Walshaw C (2001) A multilevel approach to the graph colouring problem. Tech Rep 01/IM/69, Comp Math Sci Univ, Greenwich
Xu L, Hutter F, Hoos H, Leyton-Brown K (2008) SATzilla: portfolio-based algorithm selection for SAT. J Artif Intell Res (JAIR) 32:565–606
Zhipeng L, Jin-Kao H (2012) Adaptive memory-based local search for MAX-SAT. Applied Soft Computing
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Bouhmala, N. A multilevel learning automata for MAX-SAT. Int. J. Mach. Learn. & Cyber. 6, 911–921 (2015). https://doi.org/10.1007/s13042-015-0355-4
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DOI: https://doi.org/10.1007/s13042-015-0355-4