Probabilistic Analysis of the Number Partitioning Problem
Abstract
Given a sequence of
N
positive real numbers
{
a
1
,
a
2
,...,
a
N
}
, the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of
a
j
over the two sets is minimized. In the case that the
a
j
's are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite-range, random anti-ferromagnetic Ising model. We employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large
N
limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips, and found that it vanishes like
N
−3/2
.