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Unformatted text preview: g, orders of magnitude harder than those generated by the uniform model. In Table 2 we compare the performance of DPbacktracking on uniform 3cnf problems and on 3cnf chain problems with the same number of clauses. The problems contain 25 subtheories with 5 variables and 9 to 23 3cnf clauses per subtheory, as well as 24 2cnf clauses connecting subtheories in the chain. The corresponding uniform 3cnf problems have 125 variables and 249 to 599 clauses. We tested DPbacktracking on both classes of problems. 21 Table 2 shows the mean values on 20 experiments where the number of experiments is per a constant problem size, i.e. per each row in the table. The mindiversity ordering have been used for each instance. Table 2: DPbacktracking on uniform 3cnfs and on chain problems of the same size 3cnfs theories with 125 variables Mean values on 20 experiments per each row Num of ses 249 100 299 100 349 100 399 100 449 100 499 549 599 95 35 0 Uniform 3cnfs
% Time Dead 1st solu tion 0.2 0.2 0.2 0.2 0.4 3.7 8.5 6.6 0 100 0 100 3 2 17 244 535 382 % ends Sat 3cnf chains
Time 1st solu tion 0.3 0.4 0 1 Dead ends clau Sat 70 9945.7 908861 25 2551.1 207896 15 0 0 0 185.2 2.4 0.9 0.1 13248 160 9 6 Most of the extremely hard chain problems with many deadends were found around the crossover point, where about 50% of generated chain problems were satisable. As it was shown for uniform 3cnfs [17, 4], the percentage of satisable problems and their complexity depend on the clauses/variables ratio. Small values of that ratio correspond to underconstrained problems most of which have many solutions and are easily solved by DPbacktracking. When the ratio is large, the problems become overconstrained, and mostly unsatisable. On the average, overconstrained problems are not very hard for DP22 Table 3: DR and DPbacktracking on 3cnf chains 3cnf chains with 125 variables: 25 subtheories, 5 variables in each Mean values on 20 experiments per each row Num SAT: DPbacktracking of cls 249 299 349 399 449 499 549 599 100 100 70 25 15 0 0 0 % Time: solution 0.3 0.4 9945.7 2551.1 185.2 2.4 0.9 0.1 Dead ends 0 1 908861 207896 13248 160 9 6 SAT only 0.6 1.4 2.2 2.8 3.7 3.8 4.0 4.6 0.3 0.3 0.3 0.2 0.3 0.0 0.0 0.0 solution DR
Time: 1st Time: 1st Number Size of Induced of new clauses 61 105 131 131 135 116 99 93 max clause 4.1 4.1 4.0 4.0 4.0 3.9 3.9 3.6 5.1 5.3 5.3 5.3 5.5 5.4 5.2 5.2 width backtracking, which detects inconsistency early in the search. Most of the hard problems appear around the so called crossover point, the transition point from mostly satisable to mostly unsatisable problems. According to experimental studies, for uniform 3cnfs the transition occurs at the clauses/variables ratio approximately equal to 4.3. We observed that the crossover point for chains shifted towards a smaller ratio(see Table 2). Many chain problems around the crossover point were orders of magnitude harder for DPbacktracking than uniform 3cnf problems of the same size, and also...
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This note was uploaded on 09/17/2013 for the course PMATH 330 taught by Professor W. alabama during the Spring '09 term at Waterloo.
 Spring '09
 W. Alabama

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