Subset Sum and the Distribution of Information.

The complexity of the subset sum problem does not only depend on the lack of an exact algorithm that runs in subexponential time with the number of input values. It also critically depends on the number of bits m of the typical integer in the input: a subset sum instance of n with large m has fewer solutions than a subset sum instance with relatively small m. Empirical evidence from this study suggests that this image of complexity has a more fine-grained structure. A depth-first branch and bound algorithm deployed to the integer partition problem (a special case of subset sum) shows that for this experiment, its hardest instances reside in a region where informational bits are equally dispersed among the integers. Its easiest instances reside there too, but in regions of more eccentric informational dispersion, hardness is much less volatile among instances. The boundary between these hardness regions is given by instances in which the ith element is an integer of exactly i bits. These findings show that, for this experiment, a very clear hardness classification can be made even on the basis of information dispersal, even for subset sum instances with identical values of n and m. The role of the 'scale free' region are discussed from an information theoretical perspective.