A faster FPTAS for the Unbounded Knapsack Problem

Jansen, Klaus; Kraft, Stefan Erich Julius

doi:10.1016/j.ejc.2017.07.016

Cited by 12 publications

(4 citation statements)

References 24 publications

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“…The historically first FPTAS for the 1-BKP was given in the seminal work of O. Ibarra and C. Kim [7]. The results of [7] were improved in many ways, for example in the works [2,6,5,8,10,11,12,16,18,19,21]. But, it was shown in [14] (see [13, p. 252] for a simplified proof) that the 2-BKP does not admit an FPTAS unless P = N P .…”

Section: Description Of the Results And Related Workmentioning

confidence: 99%

An FPTAS for the $Δ$-modular multidimensional knapsack problem

Gribanov

2021

Preprint

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It is known that there is no EPTAS for the m-dimensional knapsack problem unless W [1] = F P T . It is true already for the case, when m = 2. But an FPTAS still can exist for some other particular cases of the problem.In this note we show that the m-dimensional knapsack problem with a ∆-modular constraints matrix admits an FPTAS with the arithmetical complexity, where T LP is the linear programming complexity bound. In particular, for fixed m the arithmetical complexity bound becomesOur algorithm is actually a generalisation of the classical FPTAS for the 1dimensional case.The goal of the paper is only to prove the existence of an FPTAS, and more accurate analysis can give better constants in exponents. Moreover, we are not worry to much about memory usage, it can be roughly estimated as O(n • (1/ε) m+3 • (2m) 2m+6 • ∆).

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Section: Description Of the Results And Related Workmentioning

confidence: 99%

An FPTAS for the $Δ$-modular multidimensional knapsack problem

Gribanov

2021

Preprint

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“…Intuitively, ABS(p, ρ/6) computes one configuration for each layer, which is added to the solution x in the next step of the algorithm. The above integer program is equivalent to the integer program of the Unbounded Knapsack problem and therefore can be solved approximatively with approximation ratio (1 − ρ/6) in O(|J S,W | + (log(1/ρ)) 3 /ρ 2 ) operations [12]. The algorithm needs at most O(M(ln(M) + ρ −2 )) steps where it calls the ABS(p, ρ/6) exactly |S| times.…”

Section: Lemmamentioning

confidence: 99%

“…The first three constraints (12) to (14) ensure that the y-coordinates are positioned in the right order and that we use exactly the width of the strip. Furthermore, the variables w j for the width between the y-coordinates are defined.…”

Section: Positioning Containers As Well As Large and Vertical Rectanglesmentioning

confidence: 99%

Linear Time Algorithms for Multiple Cluster Scheduling and Multiple Strip Packing

Jansen

Rau

2019

Lecture Notes in Computer Science

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We study the Multiple Cluster Scheduling problem and the Multiple Strip Packing problem. For both problems, there is no algorithm with approximation ratio better than 2 unless P = NP. In this paper, we present an algorithm with approximation ratio 2 and running time O(n) for both problems. While a 2 approximation was known before, the running time of the algorithm is at least Ω(n 256 ) in the worst case. Therefore, an O(n) algorithm is surprising and the best possible. We archive this result by calling an AEPTAS with approximation guarantee (1 + ε)OPT + p max and running time of the form O(n log(1/ε) + f (1/ε)) with a constant ε to schedule the jobs on a single cluster. This schedule is then distributed on the N clusters in O(n). Moreover, this distribution technique can be applied to any variant of of Multi Cluster Scheduling for which there exists an AEPTAS with additive term p max .While the above result is strong from a theoretical point of view, it might not be very practical due to a large hidden constant caused by calling an AEPTAS with a constant ε ≥ 1/8 as subroutine. Nevertheless, we point out that the general approach of finding first a schedule on one cluster and then distributing it onto the other clusters might come in handy in practical approaches. We demonstrate this by presenting a practical algorithm with running time O(n log(n)), with out hidden constants, that is a 9/4-approximation for one third of all possible instances, i.e, all instances where the number of clusters is dividable by 3, and has an approximation ratio of at most 2.

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“…FPTASs on several special cases of 0-1 knapsack are also of interest. For the unbounded knapsack problem, where every item has infinitely many copies, Jansen and Kraft [7] obtained an O(n + ( 1 ε ) 2 log 3 1 ε )-time algorithm; the unbounded version can be reduced to 0-1 knapsack with only a logarithmic blowup in the problem size [5]. For the subset sum Kellerer and Pferschy [11] 2004 O(n log 1 ε + ( 1 ε ) 5/2 log 3 1 ε ) (randomized) Rhee [15] 2015…”

Section: Introductionmentioning

confidence: 99%

An Improved FPTAS for 0-1 Knapsack

Jin

2019

Preprint

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The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor 1 + ε runs in Õ(n + (1/ε) 12/5 ) time, where Õ hides polylogarithmic factors. In this paper we present an improved algorithm in Õ(n + (1/ε) 9/4 ) time, with only a (1/ε) 1/4 gap from the quadratic conditional lower bound based on (min, +)-convolution. Our improvement comes from a multi-level extension of Chan's number-theoretic construction, and a greedy lemma that reduces unnecessary computation spent on cheap items. ACM Subject Classification Theory of computation → Algorithm design techniquesKeywords and phrases approximation algorithms, knapsack, subset sum Acknowledgements Part of this research was done while visiting Harvard University. I would like to thank Professor Jelani Nelson for introducing this problem to me, advising this project, and giving many helpful comments on my writeup.

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A faster FPTAS for the Unbounded Knapsack Problem

Cited by 12 publications

References 24 publications

An FPTAS for the $Δ$-modular multidimensional knapsack problem

An FPTAS for the $Δ$-modular multidimensional knapsack problem

Linear Time Algorithms for Multiple Cluster Scheduling and Multiple Strip Packing

An Improved FPTAS for 0-1 Knapsack

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