Solving dense subset-sum problems by using analytical number theory
نویسندگان
چکیده
منابع مشابه
Solving Medium-Density Subset Sum Problems in Expected Polynomial Time
The subset sum problem (SSP) can be briefly stated as: given a target integer E and a set A containing n positive integer aj , find a subset of A summing to E. The density d of an SSP instance is defined by the ratio of n to m, where m is the logarithm of the largest integer within A. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme fo...
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Definition 1. Subset sum (SSUM) is the following problem: given a1, . . . , an ∈ [0, X] and s = ∑ aixi where each xi ∈ {0, 1}, find ~x = (x1, . . . , xn). Definition 2. The density of a subset sum problem is defined as n logX ; the ratio between the number of elements in your sum to the number of bits in the range of ais. Low density means n logX is very small, for example 1 n2 where X = 2 n . ...
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The Subset Sum problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O∗(2n/2)-time algorithm for Subset Sum by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a “truly faster”, O∗(2(0.5−δ)n)-time algorithm, with some constant δ > 0. Continuing an earlier wo...
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Subset sum or Knapsack problems of dimension n are known to be hardest for knapsacks of density close to 1. These problems are NP-hard for arbitrary n. One can solve such problems either by lattice basis reduction or by optimized birthday algorithms. Recently Becker, Coron, Joux [BCJ10] present a birthday algorithm that follows Schroeppel, Shamir [SS81], and HowgraveGraham, Joux [HJ10]. This al...
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ژورنال
عنوان ژورنال: Journal of Complexity
سال: 1989
ISSN: 0885-064X
DOI: 10.1016/0885-064x(89)90025-3