Hard Instances
Sets with only one, or no partitions tend to be hardest (or most expensive) to solve compared to their input sizes. When the values are small compared to the size of the set, perfect partitions are more likely. The problem is known to undergo a "phase transition"; being likely for some sets and unlikely for others. If m is the number of bits needed to express any number in the set and n is the size of the set then tends to have many solutions and tends to have few or no solutions. As n and m get larger, the probability of a perfect partition goes to 1 or 0 respectively. This was originally argued based on empiricial evidence by Gent and Walsh, then using methods from statistical physics by Mertens, and later proved by Borgs, Chayes, and Pittel.
Read more about this topic: Partition Problem
Famous quotes containing the words hard and/or instances:
“It is hard to find someone who will give your children a feeling of security while it lasts and not wound them too much when it is finished, who will treat those children as if they were her own, but knows—and never forgets—that they are yours.”
—Anna Quindlen (20th century)
“What instances must pass before them of ardent, disinterested, self-denying attachment, of heroism, fortitude, patience, resignation—of all the conflicts and the sacrifices that enno ble us most. A sick room may often furnish the worth of volumes.”
—Jane Austen (1775–1817)