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Unformatted text preview: s. Experiments with backjumping on the instances in Table 3 conrmed that it outperforms DPbacktracking by far ( Figure 11). When experimenting with BDRDP on chains we observed that all chain instances hard for DPbacktracking have been solved easily by BDRDP. The performance of BDRDP 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" DPBacktracking, DR, BDRDP and Backjumping on 3CNF chains 25 subtheories, 5 variables in each 50 experiments per each point
100000 10000 BDRDP (bound=3) DR Backjumping DPbacktracking CPU time (log scale) 1000 100 10 1 .1 240 290 340 390 440 490 540 590 640 690 Number of clauses Figure 11: DPBacktracking, DR and Backjumping on chains 26 6.3 Results on kmtrees The behaviour of both DPbacktracking 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 DPbacktracking around the (k; m)trees' crossover point. Experiments on 14trees and on 24trees, with total of 100 cliques, show that DPbacktracking exceeded the limit of 20000 deadends (around 700 seconds) on 40% of 14trees with Nclauses = 13, and on 20% of 24trees 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 dierence in CPU time between BDRDP and DPbacktracking is smaller than that on chains where we ran DPbacktracking until completion. As in the case of chains, we observed that most of the exceptionally hard problems were unsatisable. 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 DPbacktracking 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) satisable subtheories which may end up with an unsatisable subtheory causing many deadends. If at the same time m is reasonably small than DR and BDR easily recognize the unsatisable subtheory. Note that in one case a satisable instance that was relatively easy for DPbacktracking became very hard for the same algorithm after preprocessing by BDR. The problem is a 34tree with 80 clusters and 9 clauses per cluster, solved by DPbacktracking in about 3 seconds with only 58 deadends. After preprocessing, DPbacktracking 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 BDRDP is the overall superior among the algorithms considered in the paper: it is more ecient than DPBacktracking and...
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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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