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

Galil, Zvi; Margalit, Oded

doi:10.1007/3-540-54233-7_177

Cited by 19 publications

(35 citation statements)

References 9 publications

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“…We remark that the conference version of Galil and Margalit's paper [26] claims the result as stated in Theorem 1.1, but does not contain all proof details. The journal version of their paper [25] only proves a weaker result, assuming the stricter condition t mx 1/2 X Σ X /n. Nevertheless, their conference version was recently cited and used in [37].…”

Section: Reference Running Time Commentsmentioning

confidence: 99%

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On Near-Linear-Time Algorithms for Dense Subset Sum

Bringmann¹,

Wellnitz²

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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In the Subset Sum problem we are given a set of n positive integers X and a target t and are asked whether some subset of X sums to t. Natural parameters for this problem that have been studied in the literature are n and t as well as the maximum input number mxX and the sum of all input numbers ΣX . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in n. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time n O(1) .Our main question is: When can dense Subset Sum be solved in near-linear time O(n)? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters n, t, mxX , ΣX for which dense Subset Sum is in time O(n). For notational convenience we assume without loss of generality that t ≥ mxX (as larger numbers can be ignored) and t ≤ ΣX /2 (using symmetry). Then our dichotomy reads as follows:By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time O(n) if t mxX ΣX /n 2 . We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with t mxX ΣX /n 2 , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.

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Section: Reference Running Time Commentsmentioning

confidence: 99%

“…This open problem illustrates that we are far from a complete understanding of the complexity of Subset Sum with respect to the combined parameters n, t, mx X , Σ X . A line of work from around 1990 [15,16,22,25,26] suggests that this complexity is fairly complicated, as it lead to the following result.…”

Section: Introductionmentioning

confidence: 99%

On Near-Linear-Time Algorithms for Dense Subset Sum

Bringmann¹,

Wellnitz²

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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“…There can be two cases: a) D≤1: These low-density subset-sum problems are efficiently solved by reduction to a short vector in a lattice, as presented by Brickell [4]; Lagarias and Odlyzko [5]; Martello and Toth [6]; Coster et al [7]. b) D>1: These medium and high-density subset-sum problems are solvable by dynamic programming techniques or using analytical number theory, such as those presented in Chaimovich et al [8]; Galil and Margalit [9]; Flaxman and Przydatek [10]; with some of these failing to find a solution if certain bounds for n or R are not respected.…”

Section: Related Workmentioning

confidence: 99%

A Fast Heuristic Algorithm for Solving High-Density Subset-Sum Problems

Nag¹

2017

IJMSC

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The subset sum problem is to decide whether for a given set of integers A and an integer S, a possible subset of A exists such that the sum of its elements is equal to S. The problem of determining whether such a subset exists is NP-complete; which is the basis for cryptosystems of knapsack type. In this paper a fast heuristic algorithm is proposed for solving subset sum problems in pseudo-polynomial time. Extensive computational evidence suggests that the algorithm almost always finds a solution to the problem when one exists. The runtime performance of the algorithm is also analyzed.

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“…Since we require ν > 1 for a solution to the dhsp, the high density regime is more interesting for our purposes. Until recently, the best known result was a poly(k)-time algorithm for the case k > cN for some constant c [6,19]. These results are not helpful since they yield algorithms whose running times are exponential in log N .…”

Section: Relation To the Subset Sum Problemmentioning

confidence: 99%

Untitled

Bacon¹,

Childs²,

Dam³

2006

Chicago J. of Theoretical Comp. Sci.

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We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the so-called pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density ν = k/ log 2 N , where k is the number of copies of the hidden subgroup state and 2N is the order of the dihedral group. In particular, for ν < 1 the optimal measurement (and hence any measurement) identifies the hidden subgroup with a probability that is exponentially small in log N , while for ν > 1 the optimal measurement identifies the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group G for which Ω(log |G|) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we consider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden subgroup problem and average case subset sum problems. In particular, we show that an efficient quantum algorithm for a restricted version of the optimal measurement would imply an efficient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement.

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

Cited by 19 publications

References 9 publications

On Near-Linear-Time Algorithms for Dense Subset Sum

On Near-Linear-Time Algorithms for Dense Subset Sum

A Fast Heuristic Algorithm for Solving High-Density Subset-Sum Problems

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