2004 ACM Symposium on Applied Computing
POSTER ABSTRACT
Solving Weighted Max-Sat Optimization Problems Using a Taboo Scatter Search Metaheuristic Dalila BOUGHACI
Habiba Drias
University of Sciences and Technology Department of Computer Science BP.32 El-ALIA Bab-Ezzouar, 16111, Algiers, Algeria 00(213) 21 24 78 92
University of Sciences and Technology Department of Computer Science BP.32 El-ALIA Bab-Ezzouar, 16111, Algiers, Algeria 00 (213) 21 24 78 92
[email protected]
[email protected]
ABSTRACT In the last three decades, many researchers have focused on the Satisfiability problem and on many of its variants, in particular, on the Weighted Maximum Satisfiability problem (Max-W-Sat). The latter is known to be difficult to solve, due to a high number of local minima present in its search space. In this Work, we present a metaheuristic based on Taboo search (TS) procedure that makes use of the Scatter search (SS) paradigm. Our objective is to support a TS by a SS add-on to explore the influence of a population and combination strategies on the ability of generating high quality solutions.
General Terms Algorithms, Experimentation, Theory.
Keywords SAT, Max-W-Sat, Taboo Search, Scatter Search, Solution combination method, Hybrid metaheuristic.
1. INTRODUCTION The Satisfiability problem is a core problem in both computational complexity theory and artificial intelligence discipline. Its wide application to the domain of artificial intelligence in automatic reasoning and other domains as VLSI and graph theory motivates the huge interest shown for this problem. The decision version SAT was the first example of an NP-Complete problem [5]. The Satisfiability problem (SAT)[1] can be stated as. Given a collection of m clauses (C) involving n Boolean variables x1, x2 ,…………..xn. Determine whether or not there exists a truth assignment for C that satisfies the m clauses. When there is no such truth assignment, we say that the Data is contradictory. In this situation, we are interested to other variants of SAT; we can mention the Weighted MAX-SAT problems. In such problem a positive weight (Wi) is associated with each clause (Ci) and the objective consists in finding an assignment of truth values to the n variables that maximizes the sum of weights of satisfied clauses [1]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SAC’04, March 14–17, 2004, Nicosia, Cyprus. Copyright 2004 ACM 1-58113-812-1/03/04…$5.00.
The Weighted Maximum Satisfiablity problem is hard to solve; the interest focuses on the development and implementation of heuristics. Many metaheuristics have been tested for this problem. We can mention among these works, Taboo search [10], Simulated annealing [6], Genetic algorithms [4] and Scatter search [3]. In this paper, a Taboo Scatter hybrid procedure is proposed for this problem. Its algorithmic backbone is the Taboo search (TS) which is extended by a commonly shared collection of elite solutions. This collection is maintained by the Taboo search, which inputs quality solutions and is used by the Scatter search to construct combined solutions. An algorithm has been designed and tested on real-life Johnson problems. Numerical results are compared with those of other well known approaches; in particular, the GRASP.
2. A TABOO SCATTER SEARCH The last researches have proven that the evolutionary framework gives a promising foundation for the design of a hybrid metaheuristic procedure which is based on a combination of the taboo search with a population Strategy see [8]. The rest of this section is devoted to the design of a metaheuristic improvement procedure based on the desire to take advantage of the individual benefits of a single-solution oriented approach and a population oriented approach. So, our hybrid approach backbone is a basic taboo search [7] that works on single solutions by building neighborhoods from which a best admissible candidate is passed to the next iteration. In addition, the hybrid taboo search makes uses of population based strategies and maintains a collection of elite solutions. More precisely, the Taboo scatter hybrid (TS_SS) procedure starts with an initial solution generated randomly; then, a basic taboo search is started. The duration of this second phase (TS) is given by an input parameters nbiterTS corresponding to the number of iteration of the basic TS process. During the TS phase a new best solution is always deposited into the collection. Every nbiterSS iterations the algorithm calls the subroutine of the third phase, operating on the solutions in the collection. Those solutions represent the reference set in the basic scatter search [9]. They are a basis for creating new combined solutions using a combination operator. Our combination method randomly selects a position K to be the crossing point from the range [1, n]. The first K elements are copied from the one reference point while the second part is copied from the second reference point to create the new trial solution. For more information, the authors in [2] propose some interesting combination operator for permutation problems.
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After having built new combined solutions via the combination method cited above, the best solution is returned to TS to serve as an initial starting point which may be enhanced after resetting the taboo list. The algorithm terminates after a certain number of iterations.
4. CONCLUSION In this paper, we have presented a well known methaheuristic called Taboo Search. We have proposed to hybridize it with an evolutionary algorithm, in particular, with the scatter population based metaheuristic. Our objective is to explore the influence of both population and combination strategies on the ability of generating high quality solutions in single solutions oriented approaches. We can conclude that taboo search is a powerful procedure capable to organize and to direct operations of subordinate methods. We plan to improve our framework to implement a scatter taboo search meta-heuristic, in order to evaluate the influence of a single- solutions approach on a population oriented approach.
2.1 Taboo Scatter Outline Step1. Initialization - Set taboo scatter search parameters nbiter=0 /* number of TS_SS process iterations */ nbiter1=0 /* number of TS process iterations */ TL = Φ // In the beginning the Taboo list is empty // S* is the best solution with the minimum F* corresponds to S*, // F* objective function value that is F*=F(S*) Generate an arbitrary solution S Evaluate F (S), S*= S, F* = F
Table1. Solutions quality and running time results obtained by Taboo scatter Search (best of 10 executions under Pentium II, 350 MHz) and GRASP ([11])
Step 2 Iteration While (nbiter < itermax) do While (nbiter1 < nbitertTS and nbiter < itermax) do begin nbiter++ ; nbiter1++; Apply a basic TS process /*iteratively execute nbitertTS iterations using neighborhood operators*/ Add the good solution found to the collection of elite solutions to construct the reference set for the next phase
Benchmark Jnh201 Jnh202 Jnh203 Jnh205 Jnh207 Jnh208 Jnh209 Jnh210
If (nbiter = nbitertSS ) then begin SS process. Apply a TS process using the move (SWAP) to diversify the search, and add the diverse solution to the collection /* while performing the TS, execute an SS phase every nbitertSS Generate */ subset of the reference set as a basis for creating combined solutions For each subset produced, use the combination operator to produce new solutions. Improve the combined solurions; end;
Optimal 394238 394170 394199 394238 394238 394159 394238 394238
TS_ SS 394238 393961 394120 393907 394129 393824 394226 394238
tCPU 154.49 147.89 144.95 144.78 144.57 144.81 144.36 99.06
GRASP 394154 393680 393446 393890 394030 393893 393959 393950
tCPU 310.4 312.2 351.2 327.8 304.7 355.2 339.0 318.5
5. REFERENCES [1] Battiti. R, Portasi, Reactive search history based heuristics for MaxSat, journal of experimental algorithms, 1997.
[2] Campus V, Laguna M, Marti R. Context- independent scatter and tabu search for permutation problems. Document Internet, January 27, 2003.
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end; nbiter1=0; end; step 3. Print the best solution with the best cost.
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3. COMPUTATIONAL RESULTS
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The table below shows the results obtained by the Taboo Scatter Search. The maximum total number of iterations was set to itermax =2000. The basic TS phase parameter, nbiterTS, was set to 500 iterations, the population size was 10, the SS procedure was called every nbiterSS = 40 iterations. The move operator for TS intensification phase was the variable flipping. A second move operator is used in order to diversify the search consisting in permuting between two variables chosen at random. This phase is executed before calling the SS subroutine in order to create a collection of best solutions including diverse and high quality solutions. The results obtained are acceptable. In general, we can observe the superiority of the taboo scatter search in solving Weighted Max-Sat problem from the performance point of view. When SS is incorporated in TS the solutions space is better searched.
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