Solving dense subset-sum problems by using analytical number theory

Chaimovich, M.; Freiman, Gregory A.; Galil, Zvi

doi:10.1016/0885-064x(89)90025-3

Cited by 29 publications

(19 citation statements)

References 14 publications

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“…It was considerably improved by an earlier version of our paper. As a result of Freiman's more recent characterization [11], the best algorithm for computing B using analytical number theory [5] applies to density m > g.5+e but runs in time O(g 2 logg).…”

Section: Resultsmentioning

confidence: 99%

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An almost linear-time algorithm for the dense subset-sum problem

Galil

Margalit

1991

Automata, Languages and Programming

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DenseA b s t r a c t This paper describes a new approach for solving a large subproblem of the subset-sum problem. It is useful for solving other NP-hard integer programming problems. The limits and potential of this approach axe investigated.The approach yields an algorithm for solving the dense version of the subsetsum problem. It runs in time O(£log£), where l is the bound on the size of the elements. But for dense enough inputs and target numbers near the middle sum it runs in time O(m), where m is the number of elements. Consequently, it improves the previously best algorithms by at least one order of magnitude and sometimes by two.The algorithm yields a characterization of the set of subset sums as a collection of arithmetic progressions with the same difference. This characterization is derived by elementary number theoretic and algorithmic techniques. Such a characterization was first obtained by using analytic number theory and yielded inferior algorithms. I n t r o d u c t i o nThere are several ways to cope with the NP-hardness of optimization problems. One is to look for an approximate solution rather than the optimum. There is a vast literature about approximation algorithn~s and approximation schemes. For some problems there are very good approximation algorithms, for others, the problem is still NP-hea'd even ff we settle for an approximate solution. Another way to cope with complexity is to settle for the average case or to allow probabilistic algorithms. There are cases where this approach has paid off and faster algorithms have been discovered. But this has not been the case with NP-hard optimization problems.A third approach of coping with NP-hardness is to try to restrict the problem and design a polynomial-time algorithm, tIere too, there have been mixed results. Some

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Section: Resultsmentioning

confidence: 99%

“…More recently, the structure was used to derive algorithms which improved the dynmnic programming approach. The final algorithm derived using analytic number theory [5] requires O(£ ~ log t?) time.…”

Section: Introductionmentioning

confidence: 99%

An almost linear-time algorithm for the dense subset-sum problem

Galil

Margalit

1991

Automata, Languages and Programming

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“…This problem reduces to a study of solutions of linear equations of the form i s i X i = T , where X i are the numbers in the given set, s i ∈ 0 1 represents whether or not X i is included 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 [12]; see also Alon and Freiman [1], Chaimovich and Freiman [5]. We will use an analogous integral representation in our study of the integer partitioning problem.…”

Section: Introductionmentioning

confidence: 99%

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

Borgs

Chayes

Pittel

2001

Random Struct Algorithms

187

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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 → ∞, whereas 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: κ n = 1 − 2n −1 log 2 n + λ n /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 λ n tends to −∞ ∞, or λ ∈ −∞ ∞ , respectively. For λ n → −∞ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λ n → ∞, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is 2 λ n . Within the window, i.e., if λ n is bounded, we prove that the optimum discrepancy is bounded. Both for λ n → ∞ 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.

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“…This problem reduces to a study of solutions of linear equations of the form i siXi = T , where Xi are the numbers in the given set, si ∈ {0, 1} represents whether or not Xi is included 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 [12]; see also Alon and Freiman [1], Chaimovich and Freiman [6]. We will use an analogous integral representation in our study of the integer partitioning problem.…”

Section: Introductionmentioning

confidence: 98%

Sharp threshold and scaling window for the integer partitioning problem

Borgs¹,

Chayes²,

Pittel³

2001

Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing

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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 so-called 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.

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Solving dense subset-sum problems by using analytical number theory

Cited by 29 publications

References 14 publications

An almost linear-time algorithm for the dense subset-sum problem

An almost linear-time algorithm for the dense subset-sum problem

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

Sharp threshold and scaling window for the integer partitioning problem

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