In this section we take a closer look at the behavior of the search
cost, specifically, by examining how the behavior depends on whether
the problem has a solution and, if so, the number of solutions.
Figure 2: Median solution cost using dynamic backtracking for solvable (solid lines) and unsolvable (dashed and dotted lines) problems with number of variables n = 10 (black lines) and n = 20 (gray lines) as a function of number of nogoods divided by problem size, m/n. All problems were generated using the ``generate-select'' method except for the unsolvable problems shown by the dotted line, which were generated using the ``hill-climbing'' method. For problems of size 10, each point is the median of 1000 problems solved 100 times, except for unsolvable problems generated by ``generate-select'' at m/n = 3 (30 nogoods) and solvable problems at m/n = 14 (140 nogoods), which are based on 100 problems. For problems of size 20, each point is the median of 500 problems solved 100 times, except for unsolvable problems at m/n = 5 (100 nogoods) and solvable problems at m/n = 12 (240 nogoods), which are based on 15 and 35 problems, respectively. Error bars showing 95% confidence intervals are included, but in most cases are smaller than the size of the plotted points.
Figure 2 shows the median dynamic backtracking solution cost for solvable and unsolvable random CSPs generated as described above, for problems with number of variables n=10 and n=20, with domain size three. Except where specified otherwise in the figure caption, for problems of 10 variables we generated 1000 solvable and 1000 unsolvable problems for each point, and for problems of 20 variables we generated 500 solvable and 500 unsolvable problems for each point, using the ``generate-select'' method. We also generated unsolvable problems of 10 variables with 10 to 70 nogoods using the ``hill-climbing'' method. We overlap the range of problems generated by the two methods to show how the different generation methods affect search cost.
This figure clearly shows the easy-hard-easy pattern of solution cost for both solvable and unsolvable problems, for both problem sizes. The two methods of generating unsolvable problems give distinct curves: the unsolvable problems generated by the ``hill-climbing'' method are harder than those generated by the ``generate-select'' method. Nonetheless, both sets of problems show the same easy-hard-easy pattern.
Figure 3: Median solution cost for 3-coloring random graphs with 100 nodes as a function of connectivity using backtrack search with the Brelaz heuristic. The solid and dashed curves correspond to solvable and unsolvable cases respectively. These results started with 100,000 random graphs at each value of , and additional samples were generated at the extremes to produce at least 100 samples for each point. For random graphs, the crossover from mostly solvable to mostly unsolvable occurs around a connectivity of 4.5. Error bars showing 95% confidence intervals are included.
Another example with the same behavior is shown in Figure 3 for the median search cost for instances of 3-coloring of random graphs. In contrast to Figure 2, the solvable and unsolvable cases have similar median search costs near the peaks. This is because, as described above, the graph coloring searches for unsolvable cases used the symmetry with respect to permutations of the colors to avoid unnecessary search. Without this optimization, the costs for unsolvable cases would be six times greater than the values shown in the figure. Similar peaks are seen for other classes of graphs, such as connected ones, although at somewhat different values of .
These data show that both random CSPs and graph coloring problems exhibit an easy-hard-easy pattern for solvable and unsolvable problems considered separately.
A peak in search cost for solvable problems such as we observed has also been seen extensively in studies of local-repair search methods and for problems generated with a pre-specified solution [22, 14, 21]. These search methods start with some assignment to all of the variables in the problem and then attempt to adjust them until a solution is found. Generally, such methods are not systematic searches: they can never determine that a problem has no solution. Thus empirical studies of these methods are restricted to consider solvable problems and incidentally provide a useful examination of the properties of solvable problems.
Furthermore, a study of satisfiability problems with backtracking search is consistent with a peak in cost for solvable problems , but there were insufficient highly constrained solvable problems to make a definite conclusion for the behavior with many constraints.
Figure 4: Mean (solid) and median (dashed) number of solutions on a log scale as a function of the number of binary nogoods, for solvable problems with 10 variables, 3 values each, based on 1000 problems generated by the ``generate-select'' method at each multiple of 10 binary nogoods, except for 140 nogoods, which is based on 100 problems. At 0 nogoods there are = 59049 solutions. Error bars showing 95% confidence intervals are included.
How does the existence of a peak for solvable problems fit with the explanation given above? Certainly an explanation based on a transition from solvable to unsolvable problems cannot apply directly to the class of solvable problems. However, the competition between increased pruning and decreased number of solutions still applies. As shown in Figure 4, the number of solutions for solvable random CSPs of size 10 at first decreases rapidly as constraints are added but then nears its minimum value of one, giving a slower decrease. Except for the change in minimum value from 0 to 1 solution, this behavior for the number of solutions is qualitatively similar to that for the general case including both solvable and unsolvable problems. The additional constraints continue to increase the pruning of unproductive search paths. Thus the explanation given above might continue to apply but now predicts the peak will be at the point where solutions can drop no further (i.e., one solution) rather than becoming unsolvable (i.e., zero solutions).
Figure 5: Fraction of problems with at least two solutions as a function of number of nogoods divided by problem size, for problems of size 10 (black line) and size 20 (gray line). Data for problems of size 10 are based on 1000 solvable problems created by the ``generate-select'' method at each point, except for 100 solvable problems at m/n = 14 (140 nogoods). Data for problems of size 20 are based on 500 solvable problems at each point, except for 20 solvable problems at m/n = 12 (240 nogoods), also created by the ``generate-select'' method. Error bars showing 95% confidence intervals are included.
Figure 5 evaluates this idea. This figure shows how the fraction of problems with at least two solutions changes as a function of the number of nogoods divided by the problem size for random CSPs with 10 and 20 variables. For problems of size 10, the second to last solution disappears, on average, between 90 and 100 nogoods: the median number of solutions has dropped to 2 by 90 nogoods, and to 1 by 100 nogoods (Figure 4). The peak in solution cost for solvable problems is slightly lower than this, at between 80 and 90 nogoods, close to the crossover point of Figure 5 where half the solvable problems have only one solution. This is perhaps close enough to be consistent with the explanation given above. However, this relationship does not hold for problems of size 20. For this class of problems, the cost peak of solvable problems is at around 180 nogoods (m/n=9), whereas the point at which half the problems have just one solution has still not been reached by 240 nogoods (m/n=12). At 180 nogoods, the median number of solutions is 4 (mean is 10.0), and at 240 nogoods, the median is still 2 (mean is 1.83). This is inconsistent with the explanation that the cost peak for solvable problems is due to the increasing effect of pruning given no possible further decrease in number of solutions.
Since the explanation depending on a change to insolubility does not apply, and the pruning versus number of solutions explanation does not fit the data, some other factors must be at work to produce the easy-hard-easy pattern for solvable problems. We suspect the explanation is related to the idea of minimal unsolvable subproblems. A minimal unsolvable subproblem is a subproblem that is unsolvable, but for which any subset of variables and their associated constraints is solvable;  have referred to this aspect of SAT problems as the minimal unsatisfiable subset. The idea is that once a few bad choices have been made initially, such that the remainder of the problem becomes unsolvable, unsolvability is much harder to determine for some problems than for others. In particular, the more variables that are involved in a minimal unsolvable subproblem, the harder it is to determine that the subproblem is unsolvable. We make the conjecture that the cost peak for solvable problems is tied to the average size of the minimal unsolvable subproblem once a choice has been made that results in the remaining problem being unsolvable.
Figure 6: Median solution cost as a function of number of nogoods for problems of 10 variables, 3 values each, with exactly one solution, generated using the ``generate-select'' method (solid line), and by hill-climbing down to one solution starting from solvable problems with many solutions produced using ``generate-select'' (dotted line), solved using dynamic backtracking. Each point is the median of 1000 problems each solved 100 times, except for hill-climbing generated problems at 25, 30 and 35 nogoods and ``generate-select'' generated problems at 140 nogoods, of which there are 100. Error bars showing 95% confidence intervals are included.
A more interesting case is the behavior of the problems with no solutions shown in Figures 2 and 3. As a further example, Figure 6 shows the solution cost for problems with exactly one solution. This also shows a peak. These observations on problems with zero or one solution show that even with the number of solutions held constant, problems exhibit an easy-hard-easy pattern of solution cost.
According to the explanation of the transition, if the number of solutions is held constant then the increase in pruning will be the only factor, giving rise to a monotonic decrease in search cost as constraints are added. Instead, we see in Figures 2, 3 and 6 that even when the number of solutions is held fixed at zero or one, there is still a peak in solution cost, and at a smaller number of nogoods. Thus the existing explanation does not capture the full range of behaviors. Instead, it appears that there are other factors at work in producing hard problems. By focusing more closely on these factors we can hope to gain a better understanding of the structure of hard problems, which may lead to more precise predictions of search cost.
Figure 7: Comparison of median solution cost on a log scale using the same sets of unsolvable problems for chronological backtracking (black) and dynamic backtracking (gray). Dotted lines are for problems generated using the ``hill-climbing'' method, solid lines for the ``generate-select'' method. Each point is the median of 1000 problems each solved 100 times, except for the ``generate-select'' method at 30 nogoods, which is based on 100 problems. Error bars showing 95% confidence intervals are included, but are smaller than the size of the plotted points.
We also investigated the effect of algorithm on the pattern of solution cost in unsolvable problems by repeating the search of random CSPs using chronological backtrack. A comparison of chronological backtracking search with our previous dynamic backtrack search results for unsolvable problems is shown in Figure 7. In this figure, the curves for dynamic backtracking are the same as those for the unsolvable problems shown in Figure 2, except that here the cost curves are shown on a logarithmic scale. Interestingly, we do not see a peak in search cost for unsolvable problems using the less sophisticated method of chronological backtrack.
This observation raises an important point: the easy-hard-easy pattern is not a universal feature of search algorithms for problems restricted to a fixed number of solutions. This suggests that the competition between number of solutions and pruning, when it occurs, is sufficiently powerful to affect most search algorithms (very simple methods, such as generate-and-test, do not make use of pruning and show a monotonic increase in search cost as the number of solutions decreases), but that only some algorithms are able to exploit the features of weakly constrained problems with a fixed number of solutions that make them easy.
In contrast to our observations, a monotonic decrease in cost has been reported for unsolvable binary random constraint problems  and for unsolvable 3SAT problems . In the case of 3SAT, the explanation may well be choice of algorithm. Indeed,  recently found that incorporating conflict-directed backjumping and learning into the Tableau algorithm made a difference of many orders of magnitude in problem difficulty specifically for rare, ``exceptionally hard,'' unsatisfiable problems in the underconstrained region. It would be interesting to see whether the easy-hard-easy pattern for unsolvable problems would appear using their algorithm.
With respect to 's observations, the difference may be due to the range of problems generated, resulting from different problem generation methods. Smith and Dyer used a method akin to our ``random'' generation method, that is, generating problems without regard for solvability, then separating out the unsolvable ones. With this method, the ``hit rate'' for unsolvable problems in the underconstrained region is very low. It is possible that Smith and Dyer's data do not extend down to the point at which the cost of unsolvable problems begins to decrease simply because they stopped finding unsolvable problems before that point.
There are two possible reasons why we might have found unsolvable problems using random generation further into the underconstrained region, where Smith and Dyer did not. One possibility is that since we were specifically interested in unsolvable problems as far into the underconstrained region as possible, we may have spent more computational effort generating in that region. Indeed, at 40 nogoods, unsolvable problems occurred with frequency , and at 30 nogoods, with frequency . At that rate, problems at 30 nogoods took about six hours apiece to generate.
A second possibility relates to the details of the generation methods. In Smith and Dyer's random generation method, every pair of variables had exactly the same number of inconsistent value pairs between them. This implies a degree of homogeneity in the distribution of the nogoods. On the other hand, in our random generation method, each variable-value pair had an equal probability of being selected as a nogood, independent of one another. Thus it was at least possible in our generation method, though still of low likelihood, for the nogoods to occasionally clump, and to produce an unsolvable problem. This idea is discussed further in Section 5.
The difference in our observation and 's reinforces an important point: that a relatively subtle difference in generation methods can affect the class of problems generated. While the nogoods will be more or less evenly distributed on average using our generation method, they will also be clumped with some probability, whereas with Smith and Dyer's generation method, a homogeneous distribution over variable pairs is guaranteed. These types of problems could be different enough to sometimes produce different behavior.
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