Proof of the local REM conjecture for number partitioning. I: Constant energy scales (original) (raw)

Proof of the local REM conjecture for number partitioning

arXiv (Cornell University), 2005

The number partitioning problem is a classic problem of combinatorial optimization in which a set of nnn numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the nnn numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies -- corresponding to the costs of the partitions, and overlaps -- corresponding to the correlations between partitions. Although the energy levels of this model are {\em a priori} highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.

Probabilistic analysis of the number partitioning problem

Journal of Physics A: Mathematical and General, 1998

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 .

Proof of the local REM conjecture for number partitioning. II. Growing energy scales

Random Structures and Algorithms, 2009

We continue our analysis of the number partitioning problem with n weights chosen i.i.d. from some xed probability distribution with density . In Part I of this work, we established the so-called local REM conjecture of Bauke, Franz and Mertens. Namely, we showed that, as n ! 1, the suitably rescaled energy spectrum above some xed scale tends to a

Phase transition and landscape statistics of the number partitioning problem

2003

The phase transition in the number partitioning problem (NPP), i.e., the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution, is investigated by examining the energy landscape of this classic optimization problem. This is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure, termed a barrier tree, the leaves and internal nodes of which represent, respectively, the minima and the lowest energy saddles connecting those minima. Here we apply several measures of shape (balance and symmetry) as well as of branch lengths (barrier heights) to the barrier trees that result from the landscape of the NPP, aiming at identifying traces of the easy/hard transition. We find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights. Only the difficulty measure, given by the maximum value of the ratio between the barrier height and the energy surplus of local minima, succeeded in detecting traces of the phase transition in the tree. In adddition, we show that the barrier trees associated with the NPP are very similar to random trees, contrasting dramatically with trees associated with the p spin-glass and random energy models. We also examine critically a recent conjecture on the equivalence between the NPP and a truncated random energy model.

The number partition phase transition

1995

Abstract We identify a natural parameter for random number partitioning, and show that there is a rapid transition between soluble and insoluble problems at a critical value of this parameter. Hard problems are associated with this transition. As is seen in other computational problems, finite-size scaling methods developed in statistical mechanics describe the behaviour around the critical value. Such phase transition phenomena appear to be ubiquitous.

A physicist's approach to number partitioning

Theoretical Computer Science, 2001

The statistical physics approach to the number partioning problem, a classical NPhard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the "easy-to-solve" from the "hard-to-solve" phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size N to some extent corresponds to locating the minimum in an unsorted list of O(2 N) numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.

A Bose–Einstein approach to the random partitioning of an integer

Journal of Statistical Mechanics: Theory and Experiment, 2011

Consider N equally-spaced points on a circle of circumference N. Choose at random n points out of N on this circle and append clockwise an arc of integral length k to each such point. The resulting random set is made of a random number of connected components. Questions such as the evaluation of the probability of random covering and parking configurations, number and length of the gaps are addressed. They are the discrete versions of similar problems raised in the continuum. For each value of k, asymptotic results are presented when n, N both go to ∞ according to two different regimes. This model may equivalently be viewed as a random partitioning problem of N items into n recipients. A grand-canonical balls in boxes approach is also supplied, giving some insight into the multiplicities of the box filling amounts or spacings. The latter model is a k−nearest neighbor random graph with N vertices and kn edges. We shall also briefly consider the covering problem in the context of a random graph model with N vertices and n (out-degree 1) edges whose endpoints are no more bound to be neighbors.

Phase transition and finite-size scaling for the integer partitioning problem

Random Structures and Algorithms, 2001

We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of κ = m/n , we prove that the problem has a phase transition at κ = 1 , in the sense that for κ < 1 , there are many perfect partitions with probability tending to 1 as n → ∞, while for κ > 1 , there are no perfect partitions with probability tending to 1 . Moreover, we show that this transition is first-order in the sense the derivative of the so-called entropy is discontinuous at κ = 1 .