# Phase transition and finite‐size scaling for the integer partitioning problem

@article{Borgs2001PhaseTA,
title={Phase transition and finite‐size scaling for the integer partitioning problem},
author={Christian Borgs and Jennifer T. Chayes and Boris G. Pittel},
journal={Random Structures \& Algorithms},
year={2001},
volume={19}
}
• Published 1 October 2001
• Mathematics
• Random Structures & Algorithms
We consider the problem of partitioning n randomly chosen integers between 1 and 2m 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…
Phase diagram for the constrained integer partitioning problem
• Mathematics
Random Struct. Algorithms
• 2004
It is proved that the three phases of the $\kappa b$-plane can be alternatively characterized by the number of basis solutions of the associated linear programming problem, and by the fraction of these basis solutions whose $\pm 1$-valued components form optimal integer partitions of the subproblem with the corresponding weights.
Phase diagram for the constrained integer partitioning problem
• Mathematics
• 2004
We consider the problem of partitioning n integers into two subsets of given cardinalities such that the discrepancy, the absolute value of the difference of their sums, is minimized. The integers
Constrained Integer Partitions
• Mathematics
LATIN
• 2004
This work rigorously establishes this typical behavior of the optimal partition of n integers into two subsets of given cardinalities as a function of the two parameters $$\kappa :=n^{-1}{\rm log}_{2}M$$ and b:=|s|/n by proving the existence of three distinct “phases” in the κb-plane.
Proof of the local REM conjecture for number partitioning. I: Constant energy scales
• Mathematics
Random Struct. Algorithms
• 2009
The local REM conjecture is proved, asserting that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated, and the properly scaled energies converge to a Poisson process.
Lattice-based algorithms for number partitioning in the hard phase
• Mathematics, Computer Science
Discret. Optim.
• 2012
Proof of the local REM conjecture for number partitioning. II. Growing energy scales
• Mathematics
Random Struct. Algorithms
• 2009
This part analyzes the number partitioning problem for energy scales αn that grow with n, and shows that the local REM conjecture holds as long as n−1/4αn → 0, and fails if αn grows like κn 1/4 with κ > 0.
Proof of the local REM conjecture for number partitioning. II. Growing energy scales
• Mathematics
• 2005
We continue our analysis of the number partitioning problem with n weights chosen i.i.d. from some fixed probability distribution with density ρ. In Part I of this work, we established the so-called
Poisson convergence in the restricted k-partioning problem
• Mathematics
ArXiv
• 2004
This work generalizes the result to the case k > 2 in the restricted problem and shows that the vector of differences between the k sums converges to a k - 1-dimensional Poisson point process.
Revisiting the Random Subset Sum problem
• Mathematics, Computer Science
• 2022
This work presents an alternative proof for the Subset Sum Problem, with a more direct approach and resourcing to more elementary tools, in the hope of disseminating it even further.
Numerical evidence for phase transitions of NP-complete problems for instances drawn from Lévy-stable distributions
The graph ensemble presented is the first candidate, without specific graph structure built in, to generate graphs whose Hamiltonicity is intrinsically hard to determine, and is presented only as a potential graph ensemble to generate intrinsically hard graphs that are difficult to test for Hamiltonicity.

## References

SHOWING 1-10 OF 32 REFERENCES
PROBABILISTIC ANALYSIS OF THE NUMBER PARTITIONING PROBLEM
• Mathematics
• 1998
Given a sequence of N positive real numbers , the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two
The scaling window of the 2‐SAT transition
• Mathematics, Computer Science
Random Struct. Algorithms
• 2001
Using this order parameter, it is proved that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1 and the values of two other critical exponents are determined, showing that the exponents of 2-SAT are identical to those of the random graph.
Exponentially small bounds on the expected optimum of the partition and subset sum problems
In this paper we are interested in behavior of this problem when the xi are i.i.d. random variables. Under fairly general conditions, the median of the solution for the subset sum problem has been
Exponentially small bounds on the expected optimum of the partition and subset sum problems
• G. S. Lueker
• Mathematics, Computer Science
Random Struct. Algorithms
• 1998
In this paper we are interested in behavior of this problem when the xi are i.i.d. random variables. Under fairly general conditions, the median of the solution for the subset sum problem has been
Sharp threshold and scaling window for the integer partitioning problem
• Mathematics
STOC '01
• 2001
The problem of partitioning n integers chosen randomly between 1 and 2 into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized is considered and it is proved that the optimum discrepancy is bounded.
The Birth of the Giant Component
• Mathematics
Random Struct. Algorithms
• 1993
A “uniform” model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substanitially simpler than the analogous formulas for the classical random graphs of Erdos and Renyi.
Determining computational complexity from characteristic ‘phase transitions’
• Computer Science
Nature
• 1999
An analytic solution and experimental investigation of the phase transition in K -satisfiability, an archetypal NP-complete problem, is reported and the nature of these transitions may explain the differing computational costs, and suggests directions for improving the efficiency of search algorithms.
On the evolution of random graphs
• Mathematics
• 1984
(n) k edges have equal probabilities to be chosen as the next one . We shall 2 study the "evolution" of such a random graph if N is increased . In this investigation we endeavour to find what is the
Component Behavior Near the Critical Point of the Random Graph Process
• T. Luczak
• Mathematics
Random Struct. Algorithms
• 1990
It is shown that, with probability 1 o(l), when M ( n ) = n12 + s, s 3 C 2 + -a, then during a random graph process in some step M’ > M a “new” largest component will emerge, and when s3n-*+m, the largest component of G(n, M) remains largest until the very end of the process.