An Almost Linear-Time Algorithm for the Dense Subset-Sum Problem

Galil, Zvi; Margalit, Oded

doi:10.1137/0220072

Cited by 29 publications

(37 citation statements)

References 8 publications

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“…Still, it can happen that most of the items are in the sparse instance (i.e., ℓ ′ ≥ m 2 ) and we cannot use the approach from [27]. In that case we use Theorem 2.4 again, with running time O(m + γ ε 2 ) (see Lemma 5.2).…”

Section: Approximation Via Dense Subset Summentioning

confidence: 99%

“…In this section we will sketch the components of our mechanism (see Algorithm 1). The mechanism combines pseudo-polynomial Bringmann's [17] algorithm with Galil and Margalit [27] algorithm for dense instances of Subset Sum.…”

Section: A Framework For Efficient Approximationmentioning

confidence: 99%

“…6. We deliberately use O notation to hide constant factors (note that Galil and Margalit [27] algorithm requires that m > 1000 · √ l log l).…”

Section: Small Itemsmentioning

confidence: 99%

See 2 more Smart Citations

A Subquadratic Approximation Scheme for Partition

Mucha

Węgrzycki

Włodarczyk

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The subject of this paper is the time complexity of approximating Knapsack, Subset Sum, Partition, and some other related problems. The main result is an O(n + 1/ε 5/3 ) time randomized FPTAS for Partition, which is derived from a certain relaxed form of a randomized FPTAS for Subset Sum. To the best of our knowledge, this is the first NP-hard problem that has been shown to admit a subquadratic time approximation scheme, i.e., one with time complexity of O((n + 1/ε) 2−δ ) for some δ > 0. To put these developments in context, note that a quadratic FPTAS for Partition has been known for 40 years.Our main contribution lies in designing a mechanism that reduces an instance of Subset Sum to several simpler instances, each with some special structure, and keeps track of interactions between them. This allows us to combine techniques from approximation algorithms, pseudopolynomial algorithms, and additive combinatorics.We also prove several related results. Notably, we improve approximation schemes for 3SUM, (min, +)-convolution, and TreeSparsity. Finally, we argue why breaking the quadratic barrier for approximate Knapsack is unlikely by giving an Ω((n + 1/ε) 2−o(1) ) conditional lower bound.

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Section: Approximation Via Dense Subset Summentioning

confidence: 99%

Section: A Framework For Efficient Approximationmentioning

confidence: 99%

See 1 more Smart Citation

A Subquadratic Approximation Scheme for Partition

Mucha

Węgrzycki

Włodarczyk

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In general, the complexity of the k-SSP problem depends on the relationship between d and the modulus q. When q = O(poly(d)), dynamic programming solves the problem in polynomial time [9,20]. The trivial exhaustive search algorithm shows that k-SSP ∈ P when d = O(log log q).…”

Section: Introductionmentioning

confidence: 99%

“…The trivial exhaustive search algorithm shows that k-SSP ∈ P when d = O(log log q). It is known that k-SSP is NP-hard when d = (log q) c for constant c > 0, see [15,9]. An explicit formula for N k (D, b) was presented for the case of D = F q [16].…”

Section: Introductionmentioning

confidence: 99%

Moment subset sums over finite fields

Lai

Marino

Robinson

et al. 2020

Finite Fields and Their Applications

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The k-subset sum problem over finite fields is a classical NPcomplete problem. Motivated by coding theory applications, a more complex problem is the higher m-th moment k-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the mth moment k-subset sum problem over finite fields for each fixed m when the evaluation set is the image set of a monomial or Dickson polynomial of any degree n. In the classical case m = 1, this recovers previous results of Nguyen-Wang (the case m = 1, p > 2) [24] and the results of Choe-Choe (the case m = 1, p = 2) [3].

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Ternary Syndrome Decoding with Large Weight

Bricout

Chailloux

Debris-Alazard

et al. 2020

Lecture Notes in Computer Science

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The Syndrome Decoding problem is at the core of many code-based cryptosystems. In this paper, we study ternary Syndrome Decoding in large weight. This problem has been introduced in the Wave signature scheme but has never been thoroughly studied. We perform an algorithmic study of this problem which results in an update of the Wave parameters. On a more fundamental level, we show that ternary Syndrome Decoding with large weight is a really harder problem than the binary Syndrome Decoding problem, which could have several applications for the design of code-based cryptosystems.

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An Almost Linear-Time Algorithm for the Dense Subset-Sum Problem

Cited by 29 publications

References 8 publications

A Subquadratic Approximation Scheme for Partition

A Subquadratic Approximation Scheme for Partition

Moment subset sums over finite fields

Ternary Syndrome Decoding with Large Weight

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