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The crossover point for chains shifted towards a

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Unformatted text preview: harder than average uniform 3-cnf problems at their crossover point. Directional resolution, on the other hand, behaved in a tamed way on chains and was sometimes more than 1000 times faster than DP-backtracking. In Table 3 we compare DP-backtracking with DR on the same chain problems as in Table 2. A detailed illustration in Table 4 lists the results on selected hard instances (number of 23 Table 4: DR and DP-backtracking on hard chain instances (number of deadends > 5000) 3-cnf chains with 125 variables: 25 subtheories, 5 variables in each Num SAT: of cls 349 349 349 399 399 399 399 399 399 449 449 449 0 0 0 0 0 0 0 0 0 1 0 0 0 or 1 DP-backtracking Time: 1st solution Dead ends DR Time: 1st solution 1.5 2.4 1.9 3.6 3.1 3.1 3.0 2.2 2.9 5.2 3.0 3.5 41163.8 3779913 102615.3 9285160 55058.5 5105541 74.8 87.7 149.3 11877.6 841.8 655.5 2549.2 289.7 6053 7433 12301 975170 70057 47113 181504 21246 37903.3 3079997 deadends exceeds 5000) from Table 3. We see that extremely hard instances ( more than 100000 deadends), although rare, contribute most to the mean values. All the experiments reported so far used min-diversity ordering. When experimenting with di erent orderings (input ordering and min-width ordering) we observed similar results (Figure 9). We also ran a small set of experiments with the actual code of tableau [4], that implements the Davis-Putnam procedure with various heuristics. Its behaviour on chain problems was similar to the one of our implementation. Some problem instances that were hard for our version of DP-backtracking were easy for tableau, however there was a subset of instances that were extremely dicult for both algorithms. 24 3-CNF CHAINS 15 subtheories, 4 variables in each ’Initial ’ ordering 500 experiments 3-CNF CHAINS 15 subtheories, 4 variables in each Min-width ordering 100 experiments 100 100 CPU-time (log-scale) CPU-time (log scale) DP-splitting DP-elimination 10 10 1 1 .1 DP-backtracking DR .01 .1 0 5 10 15 Clauses per subtheory 20 0 5 10 15 Clauses per subtheory 20 (a) input ordering (b) min-width ordering Figure 9: DR and DP-Backtracking on chains with di erent orderings Almost all the hard chain problems (for DP-backtracking) were unsatis able. One plausible explanation is the existence of an unsatis able subtheory that is last in the ordering. If all other subtheories are satis able, then DP-backtracking will try to re-instantiate variables from the satis able subtheories each time it encounters a deadend. Figure 10 shows an example of a chain of satis able theories where an unsatis able one appears almost at the very end of ordering. Min-diversity and min-width orderings do not guarantee that we avoid a situation like that. Not knowing the structure hurts DP-backtracking. Choosing the right ordering would help but this may be hard to recognize without some preprocessing. Other variants of backtracking that are capable of exploiting the structure like backjumping[13, 9] would avoid such useless re-instantiation of variable...
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