A Faster FPTAS for the Unbounded Knapsack Problem

Jansen, Klaus; Kraft, Stefan Erich Julius

doi:10.1007/978-3-319-29516-9_23

Cited by 6 publications

(8 citation statements)

References 24 publications

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“…From [24] it follows that the O(n + 1 ε 1.99 ) approximation algorithm for Unbounded Knapsack is unlikely. This lower bounds proves the optimality of Jansen and Kraft [40] O(n+ 1 ε 2 ) FPTAS for Unbounded Knapsack. The current best FPTAS for Knapsack is burdened with time complexity of O(n + 1/ε 12/5 ) Chan [19].…”

Section: Corollary 510 (Partition With Trade-off)supporting

confidence: 59%

“…This means that a similar improvement is unlikely for Knapsack. Also, this shows that the algorithm of [40] for Unbounded Knapsack is optimal (up to polylogarithmic factors). This lower bound is relatively straightforward and follows from previous works [24,49] and was also observed in [19].…”

Section: Theorem 14mentioning

confidence: 81%

“…For the Knapsack problem Kellerer and Pferschy [45] gave an O(n min{log n, log (1/ε)} +1/ε 2 log (1/ε) min{n, 1/ε log (1/ε)}) time algorithm (note that the exponent in the parameter (n + 1/ε) is 3 here) and for Unbounded Knapsack Jansen and Kraft [40] gave an O(n + 1/ε 2 log 3 (1/ε)) time algorithm (the exponent in (n+1/ε) is 2, see Appendix C for the definition of Unbounded Knapsack). Very recently Chan [19] presented the currently best O(n + 1/ε 12/5 ) algorithm for the Knapsack.…”

Section: History Of Approximation Schemes For Knapsack-type Problemsmentioning

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: Corollary 510 (Partition With Trade-off)supporting

confidence: 59%

Section: Theorem 14mentioning

confidence: 81%

Section: History Of Approximation Schemes For Knapsack-type Problemsmentioning

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 fact, they are used to solve more general problems, i.e., 0/1 Knapsack + and Unbounded Knapsack + , where we are asked to output answers for each 0 < t ′ ≤ t. There is also a long line of research on FP-TAS for Knapsack, with the current best running times being O(n + 1 ǫ 2.4 ) for 0/1 Knapsack [16] and O(n + 1 ǫ 2 ) for Unbounded Knapsack [33].…”

Section: /1 Knapsackmentioning

confidence: 99%

On Problems Equivalent to (min,+)-Convolution

Cygan

Mucha

Węgrzycki

et al. 2019

ACM Trans. Algorithms

100

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In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower bounds -reductions from a problem assumed to be hard. The hard problems include 3SUM, All-Pairs Shortest Path, SAT, Orthogonal Vectors, and others.In the (min, +)-convolution problem, the goal is to compute a sequence (c. This can easily be done in O(n 2 ) time, but no O(n 2−ε ) algorithm is known for ε > 0. In this paper, we undertake a systematic study of the (min, +)-convolution problem as a hardness assumption.First, we establish the equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min, +)-convolution problem has been used as a building block in algorithms for many problems, notably problems in stringology. It has also appeared as an ad hoc hardness assumption. Second, we investigate some of these connections and provide new reductions and other results. We also explain why replacing this assumption with the SETH might not be possible for some problems.

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“…Later, [15] showed an FPTAS with time complexity O(n log n+ 1 ǫ 2 (n+log 1 ǫ )) and space complexity O(n+ 1 ǫ 2 ). In 2018, [14] presented an FPTAS that runs in O(n 1 ǫ 2 log 3 1 ǫ ) time and requires O(n + 1 ǫ log 2 1 ǫ ) space. However, in some applications, precisely knowing the size of each item is not realistic.…”

Section: Related Workmentioning

confidence: 99%

An FPTAS for Stochastic Unbounded Min-Knapsack Problem

Jiang,

Zhao

2019

Preprint

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In this paper, we study the stochastic unbounded min-knapsack problem (Min-SUKP). The ordinary unbounded min-knapsack problem states that: There are n types of items, and there is an infinite number of items of each type. The items of the same type have the same cost and weight. We want to choose a set of items such that the total weight is at least W and the total cost is minimized. The Min-SUKP generalizes the ordinary unbounded min-knapsack problem to the stochastic setting, where the weight of each item is a random variable following a known distribution and the items of the same type follow the same weight distribution. In Min-SUKP, different types of items may have different cost and weight distributions. In this paper, we provide an FPTAS for Min-SUKP, i.e., the approximate value our algorithm computes is at most (1 + ǫ) times the optimum, and our algorithm runs in poly(1/ǫ, n, log W ) time.

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A Faster FPTAS for the Unbounded Knapsack Problem

Cited by 6 publications

References 24 publications

A Subquadratic Approximation Scheme for Partition

A Subquadratic Approximation Scheme for Partition

On Problems Equivalent to (min,+)-Convolution

An FPTAS for Stochastic Unbounded Min-Knapsack Problem

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