Fig. 5: Perfect solutions in CNP problems.
a The probability of the existence of a perfect solution is plotted against various problem sizes N at fixed values of bias S. b The color map shows the probability of the existence of a perfect solution in a random CNP problem instance with N integers and various bias values, and the integers are chosen uniformly and randomly in the range [1, 212]. The probability at each point in the phase space is calculated over 200 random instances. Three phases are identified in the figure, separated by the orange and red dash lines. Region 1, 2, and 3 correspond to the “ordered”, “hard”, and “perfect” phases proposed in ref. 60. Region 1 in the graph is not drawn because it is likely to be trivially easy to find the optimum partition in this region for an average problem instance, so it is not meaningful to investigate the probability of a perfect solution’s existence in this region. Data used to plot (a) and (b) can be found in the data tables given in the “Figure 5a” tab of the Supplementary Data 1 file and in the Supplementary Data 6 file, respectively.
