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19
The Constrainedness of Search
 In Proceedings of AAAI96
, 1999
"... We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition ..."
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Cited by 117 (26 self)
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We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition of constrainedness generalizes a number of parameters used to study phase transition behaviour in a wide variety of problem domains. As well as predicting the location of phase transitions in solubility, constrainedness provides insight into why problems at phase transitions tend to be hard to solve. Such problems are on a constrainedness "knifeedge", and we must search deep into the problem before they look more or less soluble. Heuristics that try to get off this knifeedge as quickly as possible by, for example, minimizing the constrainedness are often very effective. We show that heuristics from a wide variety of problem domains can be seen as minimizing the constrainedness (or proxies ...
Typical random 3SAT formulae and the satisfiability threshold
 in Proceedings of the Eleventh ACMSIAM Symposium on Discrete Algorithms
, 2000
"... Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to o ..."
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Cited by 87 (2 self)
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Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to other problems, such as the 3colourability of random graphs. 1
Phase transition and finitesize scaling for the integer partitioning problem
, 2001
"... 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 ..."
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Cited by 17 (2 self)
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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 firstorder in the sense the derivative of the socalled entropy is discontinuous at κ = 1. We also determine the finitesize 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.
A physicist’s approach to number partitioning
 Theoret. Comput. Sci
, 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 “easytosolve ” from the “hardto ..."
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Cited by 10 (1 self)
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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 “easytosolve ” from the “hardtosolve ” phase of the NPP as well as for the probability distributions of the optimal and suboptimal 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 Complete Anytime Algorithm for Balanced Number Partitioning
, 1999
"... Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest dier ..."
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Cited by 8 (1 self)
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Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest dierencing method (BLDM) and Korf's complete KarmarkarKarp algorithm to get a new algorithm that optimally solves the balanced partitioning problem. For numbers with twelve signicant digits or less, the algorithm can optimally solve balanced partioning problems of arbitrary size in practice. For numbers with greater precision, it rst returns the BLDM solution, then continues to nd better solutions as time allows. Key words: Number partitioning; Anytime algorithm; NPcomplete 1 Introduction and overview The number partitioning problem is dened as follows: Given a list x 1 ; x 2 ; : : : ; x n of nonnegative, integer numbers, nd a partition A f1; : : : ; ng such that the partition di...
Sharp Threshold and Scaling Window for the Integer Partitioning Problem
, 2001
"... We consider the problem of partitioning n integers chosen randomly 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 origi ..."
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Cited by 7 (2 self)
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We consider the problem of partitioning n integers chosen randomly 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 sharp threshold 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 the derivative of the socalled entropy is discontinuous at κ = 1. We also determine the 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 0, 1, 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(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, the limiting distribution of the (scaled) discrepancy is found.
MultiWay Number Partitioning
"... The number partitioning problem is to divide a given set of integers into a collection of subsets, so that the sum of the numbers in each subset are as nearly equal as possible. While a very efficient algorithm exists for optimal twoway partitioning, it is not nearly as effective for multiway part ..."
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Cited by 5 (2 self)
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The number partitioning problem is to divide a given set of integers into a collection of subsets, so that the sum of the numbers in each subset are as nearly equal as possible. While a very efficient algorithm exists for optimal twoway partitioning, it is not nearly as effective for multiway partitioning. We develop two new linearspace algorithms for multiway partitioning, and demonstrate their performance on three, four, and fiveway partitioning. In each case, our algorithms outperform the previous state of the art by orders of magnitude, in one case by over six orders of magnitude. Empirical analysis of the running times of our algorithms strongly suggest that their asymptotic growth is less than that of previous algorithms. The key insight behind both our new algorithms is that if an optimal kway partition includes a particular subset, then optimally partitioning the numbers not in that set k−1 ways results in an optimal kway partition. 1
Phase diagram for the constrained integer partitioning problem. Random Structures Algorithms
"... 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 are i.i.d. random variables chosen uniformly from the set {1,..., M}. We study how the typical behavior ..."
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Cited by 5 (0 self)
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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 are i.i.d. random variables chosen uniformly from the set {1,..., M}. We study how the typical behavior of the optimal partition depends on n, M and the bias s, the difference between the cardinalities of the two subsets in the partition. In particular, we rigorously establish this typical behavior as a function of the two parameters κ:= n −1 log 2 M and b: = s/n by proving the existence of three distinct “phases ” in the κbplane, characterized by the value of the discrepancy and the number of optimal solutions: a “perfect phase ” with exponentially many optimal solutions with discrepancy 0 or 1; a “hard phase ” with minimal discrepancy of order Me −Θ(n) ; and a “sorted phase ” with an unique optimal partition of order Mn, obtained by putting the (s + n)/2 smallest integers in one subset. Our phase diagram covers all but a relatively small region in the κbplane. We also show that the three phases 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 ±1valued components form optimal integer partitions of the subproblem with the corresponding weights. We show in particular that this fraction is one in the sorted phase, and exponentially small in both the perfect and hard phases, and strictly exponentially smaller in the hard phase than in the perfect phase. Open problems are discussed, and numerical experiments are presented.
Enhancing the performance of memetic algorithms by using a matchingbased recombination algorithm: Results on the number partitioning problem  Results on . . .
 METAHEURISTICS: COMPUTERDECISION MAKING
, 2003
"... The Number Partitioning Problem (MNP) remains as one of the simplesttodescribe yet hardesttosolve combinatorial optimization problems. In this work we use the MNP as a surrogate for several related realworld problems, in order to test new heuristics ideas. To be precise, we study the use of we ..."
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Cited by 4 (1 self)
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The Number Partitioning Problem (MNP) remains as one of the simplesttodescribe yet hardesttosolve combinatorial optimization problems. In this work we use the MNP as a surrogate for several related realworld problems, in order to test new heuristics ideas. To be precise, we study the use of weightmatching techniques in order to devise smart memetic operators. Several options are considered and evaluated for that purpose. The positive computational results indicate that —despite the MNP may be not the best scenario for exploiting these ideas — the proposed operators can be really promising tools for dealing with more complex problems of the same family.
PROOF OF THE LOCAL REM CONJECTURE FOR NUMBER PARTITIONING I: CONSTANT ENERGY SCALES
, 2005
"... The number partitioning problem is a classic problem of combinatorial optimization in which a set of n 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 n numbers are i.i.d. variables d ..."
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Cited by 3 (0 self)
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The number partitioning problem is a classic problem of combinatorial optimization in which a set of n 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 n numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a meanfield 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 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.