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 .

Relaxation in graph coloring and satisfiability problems

“…The barriers in equation (12), which are proportional to the logarithm of the energy, do not appear in any of the coarsening classes introduced by Lai et al [54]. The difference seems to be that here there are entropy barriers.…”

“…As an example, phase transitions in optimization problems have been discovered and studied using statistical mechanics [6,7]. Other problems studied using physical methods include the knapsack-problem [8], graph-partitioning [9], minimax-games [10], the 8-Queens problem [11], number partitioning [12,13] and the stable marriage problem [14]. Field theory has also been used to study, e.g., the enumeration of Hamiltonian cycles on graphs [15] and coloring of random, planar graphs [16,17].…”

Understanding Phase Transitions with Local Optima Networks: Number Partitioning as a Case Study

“…Further studies within the physics community, have confirmed the existence of the NPP phase transition characterising it rigorously [17,18]. However, they provide no direct answer to the question of what features of the corresponding fitness landscapes, if any, are responsible for the widely different observed behaviour.…”

Heuristics and exact methods for number partitioning

“…According to [6] there is an expected exponential number of perfect partitions for κ < κc, and no expected perfect partition for κ > κc. We generated instances with several different sizes, ranging from sets A with ten elements (N = 10) to a thousand (N = 1000).…”