I. Gent and T. Walsh. An Empirical Analysis of Search in GSAT. JAIR, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... checking, the Davis Putnam procedure [2] cannot practically handle expressions with more than a few hundred of variables [10] Recently a new, very promising approach to the solution of hard satisfiability problems has been proposed which is based on greedy local search procedures (GSAT) [10, 5]. Given an expression e in CNr such as rq. 2, GSAT works as follows: 1. Randomly initialise the variables in e. 2. If e = T then return(T) 3. Select a variable such that a change in its truth assignment gives the largest increase in the total number of clauses of e that are satisfied and ....

....assignment. 4. Iterate steps 2 3 for Nfiips times. 5. Iterate steps 1 4 for Ntries times. This procedure allows finding solutions for satisfiability problems including sev eral hundred (or even thousands) of variables. Although a theoretical analysis of the the algorithm has been undertaken [5], the reason why the simple op timisation of the number of true clauses in an expression leads so frequently to finding an assignment that satisfies such an expression is actually not completely understood. PL0 provides a possible explanation for this. If ep is the PL0 version of an expression e ....

I. P. Gent and T. Walsh. An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:47 59, 1993.

....probability 1 p. It resembles simulated annealing (SA) that accepts descents with a predetermined probability and allows a certain level of ascents in variable space. WalkSAT has been very successful in solving many hard SAT problems. Some theoretical analysis of GSAT WalkSAT can be found in [101, 70]. Tabu search [75] is a method that records previously seen patterns using a simple data structure and tries to avoid those patterns in the future. A possible implementation is to maintain a tabu list in order to force a search to explore unknown unvisited regions in the variable space. Its ....

I. P. Gent and T. Walsh. An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:25--37, 1993.

....it is better to pick variables that are near the tail of the queue) This weakness of the FIFO strategy becomes increasingly significant as the problem sizes are scaled up, due to two reasons. First, as the number of variables increases, the size of the best gain bucket increases (Gent and Walsh [5] showed that the size of the best gain bucket scales linearly with the size of the problem) In addition, we showed in Section 4 that for smaller problem instances, the variable selection heuristic has a less significant impact than for large problem instances, because each variable flip has a ....

I.P. Gent and T. Walsh. An empirical analysis of search in gsat. Journal of Artificial Intelligence Research, 1:47--59, 1993.

....reactive search, history based heuristics, non oblivious functions, Tabu Search, computer implementation. c fl0000 American Mathematical Society 0000 0000 00 1.00 .25 per page 2 R. BATTITI AND M. PROTASI both discrete and continuous based greedy algorithms have been proposed, see for example [13, 14, 7, 24] and the reviews in [15] and [22] The focus of this paper is twofold: first we examine the average performance of the recently proposed non oblivious functions for local search and demonstrate that, beyond possessing a better worst case behavior [19] these functions lead to local optima of ....

....number of iterations. While the results about non oblivious functions are novel, the fact that MAX SAT is characterized by many local optima and flat plateaus and that exploring the vicinity of a local optimum for a certain period before restarting is appropriate has been observed by many authors [14, 24, 7, 8]. It is remarkable that the above combination (NOB OB) when followed by a short number of iterations (e.g. equal to 10 Theta n) reaches results that are comparable to, and in some cases better than those obtained by Hansen and Jaumard [17] In particular, NOB OB plus 10 Theta n iterations ....

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I.P. Gent and T. Walsh. An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:47--59, 1993.

....the practical point of view there has recently been a renaissance in the study of effective heuristic algorithms based on local search and random diversification techniques. Starting from the late eighties both discrete and continuous based greedy algorithms have been proposed, see for example [10, 15, 28] and the reviews in [16] and [27] MAX SAT is therefore a paradigmatic problem for the algorithmic engineering and scientific testing and tuning effort advocated for example by [2] 5] and [22] The core of this paper consists of the design of a new heuristic algorithm guided by a series of ....

I.P. Gent and T. Walsh, "An empirical analysis of search in GSAT," Journal of Artificial Intelligence Research 1 (1993), 47--59.

....the RLDs of individual instances versus a best fit exponential distribution for n = 200 test set. The horizontal lines indicate the acceptance thresholds for the 0.01 and 0.05 acceptance level. ture that the deviations are caused by the initial hill climb phase of the local search procedure (cf. [6]) Walksat (and all other algorithms used in our study) starts its search from a randomly chosen assignment, which typically violates many clauses. Consequently, the algorithm needs some time to reach the first local optimum (which possibly could be a satisfying solution) in this initial phase of ....

I.P. Gent and T. Walsh. An Empirical Analysis of Search in GSAT. Journal of Artificial Intelligence Research, 1:47--59, 1993.

....the distinct advantage that they can often provide a valid solution at any time point. This makes the technique very suitable for systems that must perform with real time guarantees. An added bonus is that the more time the hill climbing process is given, the better the solution will typically be. Gent Walsh (1993) Finally, hill climbing is especially attractive for DCSPs, because it is likely that the solution to problem C i is a good starting assignment for problem C i 1 . Freuder Wallace (1998) For these reasons, we chose to base our solution to the satellite telecommunications problem on ....

Gent, I. P. & T. Walsh (1993). Empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:47--59.

....T 0 ,T 1 , T f0 , where T i is the assignment visited after i flips have been made. We found that on Random 3 SAT with n = 100, an assignment satisfying all but a few clauses was quickly found and that during the remainder of the search, few clauses (1 10) were unsatisfied. As shown by Gent and Walsh (1993) in GSat, there is a rapid hill climbing phase, which is also suggested by Hoos (1998) followed by a long plateau like phase in which the number of unsatisfied clauses is low but constantly changing. In our experiments we used f 5 as an arbitrary indicator of the length of the hill climbing ....

Gent, I. P., & Walsh, T. (1993). An Empirical Analysis of Search in GSAT. J. Artificial Intelligence Research, 1, 47--59.

....The idea behind the parameter is that the di#culty of most of the problems should be exponential in the parameter. This supposed exponential increase in di#culty would make di#erences in the speed of the machines used to run the benchmarks relatively insignificant. For each logic, K, KT,andS4, 9 classes of formula were created, in both valid and invalid versions. For example, the branching formulae of Halpern and Moses [ 20 ] form a formula class for all three logics. Other problem classes in the set are based on the pigeon hole principle and a two colouring problem on polygons. The ....

....300 An Analysis of Empirical Testing for Modal Decision Procedures branch d4 dum grz lin path ph poly t4p K p n p n p n p n p n p n p n p n p n leanK 2. 0 1 0 1 1 0 0 0 4 2 0 3 1 2 0 0 0 #KE 13 3 13 3 4 4 3 1 2 17 5 4 3 17 0 0 3 LWB 1 0 6 7 8 6 13 19 7 13 11 8 12 10 4 8 8 11 8 7 TA 9 9 18 20 20 6 9 16 17 19 KSAT 8 8 8 5 11 17 3 4 8 5 5 13 12 10 18 SAT 1.2 12 8 12 Crack 1.0 2 1 2 3 3 1 5 2 2 6 2 3 1 1 Kris 3 3 8 6 15 13 6 9 3 11 4 5 11 7 5 FaCT 1 . 2 6 4 8 7 6 6 7 DLP 3.1 19 13 ....

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I. P. Gent and T. Walsh. An empirical analysis of search in GSAT. Journal of Artificial Inteligence Research, 1:47--57, 1993.

.... domains (type 3) For type 1, we decided to focus on sets of Random 3 SAT instances from the solubility phase transition, as these are known to be very hard for both systematic and SLS based algorithms [10, 47, 64] and they have played a prominent role in the majority of the studies in literature [55, 60, 59, 22]. For type 2, we selected sets of graph colouring problems from randomised distributions of 3 colourable graphs [54, 35] Finally, for type 3 we chose SAT encoded planning instances which have been used in a number of previous studies, as well as a subset of the DIMACS Satisfiability Benchmark ....

....for SAT are typically incomplete, i.e. even for satisfiable problem instances they cannot be guaranteed to find a solution. The reason for this is the non systematic nature of the search. Furthermore, SLS algorithms can get trapped in local minima and plateau regions of the search space [22, 15], leading to premature stagnation of the search. One of the simplest mechanisms for avoiding premature stagnation of the search is random restart, which reinitialises the search if after a fixed number of steps (cutoff time) no solution has been found. Random restart is used in almost every SLS ....

[Article contains additional citation context not shown here]

I.P. Gent and T. Walsh. An Empirical Analysis of Search in GSAT. Journal of Artificial Intelligence Research, 1:47--59, 1993.

....possible satisfying assignments may be of interest. Much research has been done in creating algorithms for solving CSPs. These mechanisms include global [Kondrak and van Beek, 1995] and local search based methods (see for example [Sosic and Gu, 1990] Minton et al. 1992] Yang and Fong, 1992] [Gent and Walsh, 1993], Gu, 1992] Selman et al. 1994] Selman and Kautz, 1993] and [Minton et al. 1990] constraint propagation problem simplifications (see for example [Nadel, 1989] Dechter, 1992] and [Prosser, 1993] hierarchical (or partial) exploitation of known problem structure (see for example ....

....many others. An understanding of both the suitability of particular algorithms to certain CHAPTER 10. CONCLUSIONS 294 problem classes, and recognition of hard classes of given problem instances is being accumulated (see [Cheeseman et al. 1991] Smith and Grant, 1995] Gent and Walsh, 1994] [Gent and Walsh, 1993], Hogg and Williams, 1994] Crawford and Auton, 1993] and [Mitchell et al. 1992] The problem of partially understanding source code through generating correspondences to a hierarchical program plan library requires the ability to create multi level mappings between code and hierarchy. ....

I.P. Gent and T. Walsh. An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:47--59, 1993. BIBLIOGRAPHY 307

....nothing in NP completeness theory or even complexity theory as a whole seems to warrant these judgments. Beyond this is the obvious vagueness in the notion of a characteristically hard problem class. This makes it difficult to design experiments to check the theory s predictions. Local Search Gent and Walsh s (1993) recent study of a local search heuristic for the satisfiability problem has some elements of empirical science. The satisfiability problem is to determine whether a given set of formulas of propositional 12 logic can be true simultaneously. An example might be, x 1 or not x 2 not x 1 or x 3 ....

Gent, I. P., and T. Walsh. 1993. An empirical analysis of search in GSAT, manuscript, Artificial Intelligence Dept., University of Edinburgh.

....We then may directly use the amount of decrease caused by a flip as its priority. We only have to extend our MFN slightly in such a way, that adjacent units with the same priority (a case that cannot occur in the basic algorithm) again have to compete for a flip. Other variants of GSAT like ISAT [6], that do not perform steepest descent but are indifferent to the slope of the descent can be parallelized by means of a MFN. As was shown in [10] the variant I 2 SAT does not perform worse than the original GSAT, but does not require a global control to select a unit to flip. Therefore, these ....

I. P. Gent and T. Walsh. An empirical analysis of search in gsat. (Electronic) Journal of Artificial Intelligence Research, 1:47--59, 1993. 13

....it is better to pick variables that are near the tail of the queue) This weakness of the FIFO strategy becomes increasingly significant as the problem sizes are scaled up, due to two reasons. First, as the number of variables increases, the size of the best gain bucket increases (Gent and Walsh [5] showed that the size of the best gain bucket scales linearly with the size of the problem) In addition, we showed in Section 4 that for smaller problem instances, the variable selection heuristic has a less significant impact than for large problem instances, because each variable flip has a ....

I.P. Gent and T. Walsh. An empirical analysis of search in gsat. Journal of Artificial Intelligence Research, 1:47--59, 1993.

....the distinct advantage that they can often provide a valid solution at any time point. This makes the technique very suitable for systems that must perform with real time guarantees. An added bonus is that the more time the hill climbing process is given, the better the solution will typically be. Gent Walsh (1993) Finally, hill climbing is especially attractive for DCSPs, because it is likely that the solution to problem C i is a good starting assignment for problem C i 1 . Freuder Wallace (1998) For these reasons, we chose to base our solution to the satellite telecommunications problem on ....

Gent, I. P. & T. Walsh (1993). Empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:47--59.

....were able to conclude that exploring new corners of the cube is not that important. This increased our interest in studying further reasons for the performance advantage of HSAT over GSAT. We focused our attention on poss flips the number of equally good flips between which GSAT randomly picks [4], or, alternatively, the branching factor of GSAT search during the plateau phase. We noticed that on earlier stages of the plateau phase both GSAT and NRGSAT tend to increase poss flips, whereas HSAT randomly oscillates poss flips around a certain (lower) level. To confirm the importance of ....

Gent, I., Walsh, T.: An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research 1 (1993) 47--59

....Simple scaling laws are often associated with these phase transitions. For example, scaling laws have been observed both for properties of random problems like the probability of having a solution [12, 7] and for properties of algorithms like the fitness of solutions during hill climbing [3]. Such scaling laws are likely to prove useful in theoretical analyses of problem hardness and algorithm performance. In this paper, we show how phase transition phenomena are of practical use in studying a typical OR problem, the traveling salesman problem (tsp) We start by showing that, ....

Ian P. Gent and Toby Walsh. An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:47--59, September 1993.

....T 0 ; T 1 ; T f0 , where T i is the assignment visited after i flips have been made. We found that on Random 3 SAT with n = 100, an assignment satisfying all but a few clauses was quickly found and that during the remainder of the search, few clauses (1 10) were unsatisfied. As shown by Gent and Walsh (1993) in GSat, there is a rapid hill climbing phase, which is also suggested by Hoos (1998) followed by a long plateau like phase in which the number of unsatisfied clauses is low but constantly changing. In our experiments we used f 5 as an arbitrary indicator of the length of the hill climbing ....

Gent, I. P., & Walsh, T. (1993). An Empirical Analysis of Search in GSAT. J. Artificial Intelligence Research, 1, 47--59.

....T 0 ; T 1 ; T f 0 , where T i is the assignment visited after i flips have been made. We found that on Random 3 SAT with n = 100, an assignment satisfying all but a few clauses was quickly found and that during the remainder of the search, few clauses (1 10) were unsatisfied. As in GSat (Gent Walsh, 1993), there is a rapid hill climbing phase, which is also suggested by Hoos (1998) followed by a long plateau like phase in which the number of unsatisfied clauses is low but constantly changing. In our experiments, we used f 5 as am arbitrary indicator of the length of the hill climbing phase. ....

Gent, I. P., & Walsh, T. (1993). An Empirical Analysis of Search in GSAT. J. Artificial Intelligence Research, 1, 47--59.

....flipped to maximize number of satisfied clauses end end return no satisfying assignment found Figure 6: The Gsat local search procedure increase the score, a variable is flipped which does not change the score. Without such sideways flips, the performance of Gsat degrades greatly. In [32] it is shown that much of the search consists of the exploration of large plateaus where sideways flips predominate and only the occasional up flip is possible. Local search procedures like Gsat s need to be implemented with some care. Naively, one simply iterates through the variables, and for ....

I.P. Gent and T. Walsh. An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:23--57, 1993.

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I. Gent and T. Walsh. An Empirical Analysis of Search in GSAT. JAIR, 1993.

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Gent, I. P. and Walsh, T. (1993). An empirical analysis of search in GSAT. Journal of Artificial Intelligence Research, 1:47--59.

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Gent, I. and Walsh, T. (1993) An empirical analysis of search in GSAT. J. of Artificial Intelligence Research, vol. 1, 1993.

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Gent, I. P., and Walsh, T. 1993a. An Empirical Analysis of Search in GSAT. (Electronic) Journal of Artificial Intelligence Research 1:47--59.

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Gent, I. and Walsh, T. (1993) An empirical analysis of search in GSAT. J. of Artificial Intelligence Research, vol. 1, 1993.

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