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Too long 62 results on chains the behaviour of the

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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 DP-backtracking on uniform 3-cnf problems and on 3-cnf chain problems with the same number of clauses. The problems contain 25 subtheories with 5 variables and 9 to 23 3-cnf clauses per subtheory, as well as 24 2-cnf clauses connecting subtheories in the chain. The corresponding uniform 3-cnf problems have 125 variables and 249 to 599 clauses. We tested DP-backtracking 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 min-diversity ordering have been used for each instance. Table 2: DP-backtracking on uniform 3-cnfs and on chain problems of the same size 3-cnfs 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 3-cnfs % 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 3-cnf 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 cross-over point, where about 50% of generated chain problems were satis able. As it was shown for uniform 3-cnfs [17, 4], the percentage of satis able 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 DP-backtracking. When the ratio is large, the problems become overconstrained, and mostly unsatis able. On the average, overconstrained problems are not very hard for DP22 Table 3: DR and DP-backtracking on 3-cnf chains 3-cnf chains with 125 variables: 25 subtheories, 5 variables in each Mean values on 20 experiments per each row Num SAT: DP-backtracking 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 cross-over point, the transition point from mostly satis able to mostly unsatis able problems. According to experimental studies, for uniform 3-cnfs 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 DP-backtracking than uniform 3-cnf 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.

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