Clauses and the induced width the number of

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Unformatted text preview: ories. In all these cases DP-backtracking signi cantly outperforms DR. It was observed that the complexity of DR indeed grows exponentially with the size of problems (see Figure 19 DR vs. DP-backtracking Uniform 3-CNF, 20 variables 20 experiments per each point 1000 DP-backtracking DR CPU time 50 BDR-DP vs. DP-backtracking Uniform 3-CNF, 150 variables 40 experiments per each point BDR-DP (bound = 3 ) DP-Backtracking CPU time (log scale) 100 40 10 30 1 20 .1 10 .01 20 40 60 80 100 120 0 500 550 600 650 700 750 Number of clauses Number of clauses (a) DR and DP-backtracking (b) BDR-DP and DP-backtracking Figure 8: DP-backtracking, DR and BDR on uniform 3-cnfs 8a). We show the results for 3-cnfs with 20 variables only. On larger problems DR often ran out of memory due to the large number of generated clauses. Since DR was so inecient for solving uniform k -cnfs we used Bounded Directional Resolution (BDR) followed by DP-backtracking (BDR-DP) using di erent bounds. Our experiments show that BDR generates almost no new clauses when running on uniform k -cnf theories with a bound k or less. On the other hand, when the bound is strictly greater than k , running just BDR takes too long. The only promising case occurs when the bound equals k . In this case relatively few clauses were added by BDR which therefore ran much faster. DP-backtracking often ran a little faster on the BDR's output theory than on the original theory, and, therefore, the combined algorithm was somewhat more ecient than DP-backtracking (see Figure 8b). Our experiments have shown that on larger problems BDR-DP becomes more ecient relatively to DP-backtracking. BDR-DP with bound 3 was also more ecient than DP-backtracking on random 3-cnfs when varying the probability of a literal to appear positive in a clause. The results of running BDR with bounds 3 and 4 on random cnfs with p = 0:7 are shown in Table 1. Again, as in the case when p = 0:5, we can see that BDR-DP with bound 3 is almost 20 Table 1: DP-backtracking (DPB) versus BDR-DP with bounds 3 and 4 on uniform 3-cnfs 200 variables Probability of a literal to appear positive = 0.7 Mean values on 20 experiments per each row Num of cls 900 1000 1100 1.1 2.7 8.8. DPB: ends 0 48 199 3688 5027 3040 BDR-DP (bound 3) DP time 1.1 1.6 27.7 time 0.3 0.4 0.6 ends 0 14 685 3271 4682 2783 of new cls 11 12 18 23 28 34 BDR-DP (bound 4) DP time 1.7 2.7 50.4 Dead Number ends 1 21 729 2711 4000 2330 of new cls 8.4 13.1 20.0 657 888 1184 1512 1895 2332 time Time Dead BDR Dead Number BDR 1200 160.2 1300 235.3 1400 155.0 0.8 141.5 1.0 219.1 1.2 142.9 28.6 225.7 39.7 374.4 54.4 259.0 always more ecient that DP-Backtracking. Running BDR with larger bounds does not look promising, because even for bound 4 the preprocessing phase takes too long. 6.2 Results on chains The behaviour of the algorithms on chains ( and k -tree embeddings in general) di ers dramatically from that on uniform instances. We found here extremely hard instances for DP-backtrackin...
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