Journal of Statistical Mechanics: Theory and Experiment
Volume 2007
October 2007
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John Ardelius et al J. Stat. Mech. (2007) P10012 doi:10.1088/1742-5468/2007/10/P10012
Clustering of solutions in hard satisfiability problems
John Ardelius1, Erik Aurell2 and Supriya Krishnamurthy3
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| Figure 1. The mean value and the variance of the overlaps between a set of 40 solutions found by ASAT. Ranging from α = 3.7 to 4.3. N = 2000 variables. |
Figure 2. Rank plot of the values of the overlap matrix
D for α = 3.5, 3.6, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.25 and 4.3, N = 2000. Self-overlaps are left out. For low values of α, the curves are continuously distributed around one mean value. When the number of constraints increases the distribution tends to spread out. Increasing
α even further results in a break-up of the distribution. The solutions found are then grouped together in clusters. The overlap of solutions in the same cluster is high
( > 0.9N), while the overlap between solutions in different clusters is lower ( ). |
| Figure 3. The result of the clustering algorithm shows how many disks with given radius are required to cover the whole set of solutions. The three curves shows this for α = 4.2, 4.25 and 4.3, N = 2000. The small subplot is an enlargement of the curve for α = 4.3. |
| Figure 4. The first plot shows overlap between a solution found by SP and one found by ASAT for α = 0 to 4.3. The other plot show the overlaps between two different solutions found by ASAT on the same instance. The number of variables are 2000 and 10 000 in both cases. |
Figure 5. (a) For each chain the point in α where the smallest overlap in the overlap matrix is above
0.8N is marked for N = 100, 200, 400, 1000 and 2000 variables. 110 chains are used for each
N. The point of the jump of each chain marks one point in the rank plot. The mean value of the
N = 2000 curve is 4.245. (b) Finite size scaling applied to the same data. The best fit is achieved for
ν = 1.7 and  . The ruggedness of the curves for large
N is due to discretization in α. |
| Figure 6. The fraction of instances that display a condensation transition within the time cut-off. |
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