## Phase transition and finite-size scaling for the integer partitioning problem (2001)

### Cached

### Download Links

Citations: | 17 - 2 self |

### BibTeX

@MISC{Borgs01phasetransition,

author = {Christian Borgs and Jennifer Chayes and Boris Pittel},

title = {Phase transition and finite-size scaling for the integer partitioning problem},

year = {2001}

}

### OpenURL

### Abstract

Dedicated to D. E. Knuth on the occasion of his 64th birthday. Abstract. 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. We also determine the finite-size scaling window about the transition point: κn = 1 − (2n) −1 log 2 n + λn/n, by showing that the probability of a perfect partition tends to 1, 0, or some explicitly computable p(λ) ∈ (0, 1), depending on whether λn tends to −∞, ∞, or λ ∈ (−∞, ∞), respectively. For λn → − ∞ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λn → ∞, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is Θ(2 λn). Within the window, i.e., if |λn | is bounded, we prove that the optimum discrepancy is bounded. Both for λn → ∞ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the k smallest discrepancies above the scaling window.

### Citations

1842 | On the evolution of random graphs - Erdős, Rényi - 1960 |

1789 | Random Graphs - BOLLOBAS - 2001 |

401 | Bounds on multiprocessing timing anomalies
- Graham
- 1969
(Show Context)
Citation Context ...the “multi-way” partition problem in which a set of “weights” is to be partitioned into N ≥ 3 subsets (parts), so that the sums of the weights in the N parts are as close to equal as possible. Graham =-=[Gra]-=- developed a linear-time 4 3 -approximation algorithm for a version of this problem in which the goal is to minimize the weight of the heaviest part. The multi-way problem was also considered by [KKLO... |

173 | Determining computational complexity from characteristic phase transitions - Monasson, Zecchina, et al. - 1999 |

67 |
The differencing method of set partitioning
- Karmarkar, Karp
(Show Context)
Citation Context ...numbers drawn from a compact interval in R, which we will henceforth refer to as the continuous case. This is conceptually analogous to the integer partitioning problem with m ≫ n. Karmarkar and Karp =-=[KK]-=- gave a linear time algorithm for a suboptimal solution with a typical discrepancy of size O(n −c log2 n ) for some constant c > 0; see [Yak] for a proof of the KK-conjecture that the discrepancy obta... |

49 | Component behavior near the critical point of the random graph process, Random Structures Algorithms - Luczak |

31 | The birth of the giant component - Janson, Knuth, et al. - 1993 |

27 |
Probabilistic analysis of optimum partitioning
- Karmakar, Karp, et al.
- 1986
(Show Context)
Citation Context ...e [Yak] for a proof of the KK-conjecture that the discrepancy obtained by their algorithm is indeed of the order n −θ(log2 n) . The optimum solution was studied by Karmarkar, Karp, Lueker and Odlyzko =-=[KKLO]-=- who proved that the typical minimum discrepancy is much smaller, namely of order O(2 −n√ n). More recently, Lueker [Lue] proved exponential bounds for the expected minimum discrepancy. Note that all ... |

24 | The scaling window of the 2-sat transition
- Bollobas, Borgs, et al.
(Show Context)
Citation Context ...rgest connected component changes from order log n to order n. More recently, there has been much study of the phase transition in the random k -SAT model, both by heuristic and rigorous methods; see =-=[BBCKW]-=- and references therein. In k -SAT, the instances are formulas in conjunctive normal form; each formula has m clauses, and each clause has k distinct literals drawn uniformly at random from among n Bo... |

24 |
Theory and Examples, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software
- Durrett, Probability
- 1991
(Show Context)
Citation Context ... to show that Zn is Gaussian in the limit. To this end, let us look close at M2 = �n j=1 X2 j . Since �n j=1 E(X2 j )3 � Var ��n j=1 X2 � j �3/2 = O(n−1/2 ), we have by Lyapunov’s theorem (see, e.g., =-=[Dur]-=-) that M2 = n� X 2 j = nE(X 2 ) + � nVar(X2 )Nn, (4.16) j=1 where Nn is standard normal in the limit, that is lim n→∞ P�Nn ≤ x � = 1 (2π) 1/2 �x −∞ e −u2 /2 du. We want to show that Nn remains standar... |

21 |
Phase transition in number partitioning problem
- Mertens
- 1998
(Show Context)
Citation Context ...tending to infinity in the limiting ratio κ, the probability of a perfect partition tends to 1 for κ < 1, while the probability tends to 0 for κ > 1. Our result was inspired by the results of Mertens =-=[Mer1]-=- who gave an ingenious, nonrigorous argument for the existence of a phase transition in optimum partitioning. We also derive the finite-size scaling of the system about the transition point κ = 1.sInt... |

16 |
The differencing algorithm LDM for partitioning: a proof of a conjecture of Karmarkar and
- Yakir
- 1996
(Show Context)
Citation Context ...integer partitioning problem with m ≫ n. Karmarkar and Karp [KK] gave a linear time algorithm for a suboptimal solution with a typical discrepancy of size O(n −c log2 n ) for some constant c > 0; see =-=[Yak]-=- for a proof of the KK-conjecture that the discrepancy obtained by their algorithm is indeed of the order n −θ(log2 n) . The optimum solution was studied by Karmarkar, Karp, Lueker and Odlyzko [KKLO] ... |

15 | Succinct certificates for almost all subset sum problems - Furst, Kannan - 1989 |

10 | Probabilistic analysis of the number partitioning problem
- Ferreira, Fontanari
- 1998
(Show Context)
Citation Context ... at a transition point estimated to be close to κ = 0.96. Ferreira and Fontanari studied the random spin model of Fu, and used statistical mechanical methods to get estimates of the optimum partition =-=[FF1]-=-s4 C. Borgs, B. Pittel, J.T. Chayes, August 2001 and to evaluate the average performance of simple heuristics [FF2]. Our work was motivated by the beautiful paper of Mertens [Mer1], who used statistic... |

10 | Exponentially Small Bounds on the Expected Optimum of the Partition and Subset Sum Problems. Random Structures and Algorithms
- Lueker
- 1998
(Show Context)
Citation Context ... n) . The optimum solution was studied by Karmarkar, Karp, Lueker and Odlyzko [KKLO] who proved that the typical minimum discrepancy is much smaller, namely of order O(2 −n√ n). More recently, Lueker =-=[Lue]-=- proved exponential bounds for the expected minimum discrepancy. Note that all of these results correspond to m ≫ n, and hence κ → ∞, well above the phase transition studied here. There have also been... |

7 | Sharp threshold and scaling window for the integer partitioning problem
- Borgs, Chayes, et al.
- 2001
(Show Context)
Citation Context ...ion of his 64th birthday. His work has been an inspiration to us throughout the years. We use this occasion to provide complete proofs and to give several extensions of the results first announced in =-=[BCP]-=-. There has recently been much interest in the study of phase transitions in random combinatorial problems. A combinatorial phase transition is an abrupt change in the qualitative behavior of the prob... |

5 |
Solving dense subset-sum problems by using analytical number theory
- Chaimovich, Freiman
- 1989
(Show Context)
Citation Context ...ea is to express the total number of solutions to these equations via a Fourier-type inversion integral, a paradigm championed by Freiman [Fre]; see also Alon and Freiman [AF], Chaimovich and Freiman =-=[CF]-=-. We will use an analogous integral representation in our study of the integer partitioning problem. Some of the methods and results presented here can be used to obtain stronger results for the subse... |

4 |
On sums of subsets of a set of integers, Combinatorica 8
- Alon, Freiman
- 1988
(Show Context)
Citation Context ...s the target number. A key idea is to express the total number of solutions to these equations via a Fourier-type inversion integral, a paradigm championed by Freiman [Fre]; see also Alon and Freiman =-=[AF]-=-, Chaimovich and Freiman [CF]. We will use an analogous integral representation in our study of the integer partitioning problem. Some of the methods and results presented here can be used to obtain s... |

4 | Statistical mechanics analysis of the continuous number partitioning problem
- Ferreira, Fontanari
- 1999
(Show Context)
Citation Context ...u, and used statistical mechanical methods to get estimates of the optimum partition [FF1]s4 C. Borgs, B. Pittel, J.T. Chayes, August 2001 and to evaluate the average performance of simple heuristics =-=[FF2]-=-. Our work was motivated by the beautiful paper of Mertens [Mer1], who used statistical mechanical methods and the parameterization of Gent and Walsh to derive a compelling argument for a phase transi... |

4 |
thresholds of graph properties and the k -sat problem
- Friedgut, Sharp
- 1999
(Show Context)
Citation Context ...oolean variables and their negations. For fixed k ≥ 2, the model undergoes a sharp transition from solvability to insolvability as the parameter α = m/n passes through a particular k -dependent value =-=[Fri]-=-. In the language of mathematical physics, phase transitions occur only in the so-called thermodynamic limit, that is, in the limit of an infinite system. Finite-size scaling describes the approach of... |

3 |
An analytical method of analysis of linear boolean equations
- Freiman
- 1980
(Show Context)
Citation Context ...d in a particular subset, and T is the target number. A key idea is to express the total number of solutions to these equations via a Fourier-type inversion integral, a paradigm championed by Freiman =-=[Fre]-=-; see also Alon and Freiman [AF], Chaimovich and Freiman [CF]. We will use an analogous integral representation in our study of the integer partitioning problem. Some of the methods and results presen... |

2 |
Random Costs in
- Mertens
- 2000
(Show Context)
Citation Context ... beautiful paper of Mertens [Mer1], who used statistical mechanical methods and the parameterization of Gent and Walsh to derive a compelling argument for a phase transition. In a later work, Mertens =-=[Mer2]-=- analyzed Fu’s model by heuristically approximating it in terms of Derrida’s random energy model [Der], and thereby obtained the limiting distribution of the kth smallest discrepancy. Merten’s random ... |

1 |
Phase transition in the number partitioning problem
- Alg
- 1998
(Show Context)
Citation Context ...tending to infinity in the limiting ratio κ, the probability of a perfect partition tends to 1 for κ < 1, while the probability tends to 0 for κ > 1. Our result was inspired by the results of Mertens =-=[Mer1]-=- who gave an ingenious, nonrigorous argument for the existence of a phase transition in optimum partitioning. We also derive the finite-size scaling of the system about the transition point κ = 1.Int... |