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Unformatted text preview: s. Experiments with backjumping on the instances in Table 3 con rmed that it outperforms DP-backtracking by far ( Figure 11). When experimenting with BDR-DP on chains we observed that all chain instances hard for DP-backtracking have been solved easily by BDR-DP. The performance of BDR-DP on chains was comparable to DR. 25 sat=1 sat=1 sat=1 sat=1 sat= 0 Figure 10: Illustration of a \hard chain problem" DP-Backtracking, DR, BDR-DP and Backjumping on 3-CNF chains 25 subtheories, 5 variables in each 50 experiments per each point 100000 10000 BDR-DP (bound=3) DR Backjumping DP-backtracking CPU time (log scale) 1000 100 10 1 .1 240 290 340 390 440 490 540 590 640 690 Number of clauses Figure 11: DP-Backtracking, DR and Backjumping on chains 26 6.3 Results on k-m-trees The behaviour of both DP-backtracking and DR on (k; m)-trees is similar to the one we have observed on chains. Indeed, chains are (2; m)-trees. For xed k , m and the number of cliques in a (k; m)-tree, we varied the number of clauses per clique and discovered, again, exceptionally hard problems for DP-backtracking around the (k; m)-trees' crossover point. Experiments on 1-4-trees and on 2-4-trees, with total of 100 cliques, show that DPbacktracking exceeded the limit of 20000 deadends (around 700 seconds) on 40% of 1-4-trees with Nclauses = 13, and on 20% of 2-4-trees with Nclauses = 12. Table 5 summarizes the results on (k; m)-trees. We terminate the algorithm once it reaches more than 20000 deadends. This explains why the di erence in CPU time between BDR-DP and DP-backtracking is smaller than that on chains where we ran DP-backtracking until completion. As in the case of chains, we observed that most of the exceptionally hard problems were unsatis able. The frequency of those hard instances and their hardness depend on the parameters of (k; m)-trees. For xed m, when k is small, and the number of cliques is large, hard instances for DP-backtracking appear more often. The intuitive explanation is similar to one given for chains: we are likely to encounter a long sequence of almost independent (because the intersection size between cliques, k , is small) satis able subtheories which may end up with an unsatis able subtheory causing many deadends. If at the same time m is reasonably small than DR and BDR easily recognize the unsatis able subtheory. Note that in one case a satis able instance that was relatively easy for DP-backtracking became very hard for the same algorithm after preprocessing by BDR. The problem is a 3-4tree with 80 clusters and 9 clauses per cluster, solved by DP-backtracking in about 3 seconds with only 58 deadends. After preprocessing, DP-backtracking encountered 10000 deadends and was terminated without nding a solution after 227 seconds. It means that preprocessing (BDR) could hurt backtracking (DP) in some rare cases. Still, we concluded that BDR-DP is the overall superior among the algorithms considered in the paper: it is more ecient than DP-Backtracking and...
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