An Improved Knapsack Solver for Column Generation

Jansen, Klaus; Kraft, Stefan Erich Julius

doi:10.1007/978-3-642-38536-0_2

“…As stated in [21] and [11,Lemma 5], non-dominated tuples (p, s, k) can be removed in linear time in the number of tuples if the tuples are different and sorted by profit. This is the case because every tuple in D (k) is stored in an array sorted according to the corresponding ξ.…”

Section: Corollary 26mentioning

confidence: 99%

A Faster FPTAS for the Unbounded Knapsack Problem

Jansen

¹

,

Kraft

²

2016

Lecture Notes in Computer Science

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The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0-1 Knapsack Problem (0-1 KP). In contrast to 0-1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time approximation schemes (FPTAS) are of great interest. Such algorithms find a solution arbitrarily close to the optimum OPT(I), i.e. of value at least (1 − ε)OPT(I) for ε > 0, and have a running time polynomial in the input length and 1 ε . For over thirty years, the best FPTAS was due to Lawler with a running time in O(n + 1 ε 3 ) and a space complexity in O(n + 1 ε 2 ), where n is the number of knapsack items. We present an improved FPTAS with a running time in O(n + 1 ε 2 log 3 1 ε ) and a space bound in O(n + 1 ε log 2 1 ε ). This directly improves the running time of the fastest known approximation schemes for Bin Packing and Strip Packing, which have to approximately solve UKP instances as subproblems. * Research supported by DFG project JA612/14-2, "Entwicklung und Analyse von effizienten polynomiellen Approximationsschemata für Scheduling-und verwandte Optimierungsprobleme" arXiv:1504.04650v2 [cs.DS] 9 Nov 2015

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“…We [20] have extended our algorithm to the Unbounded Knapsack Profit with Inversely Proportional Profits (UKPIP) introduced in [11]. Here, several knapsack sizes 0 < c 1 < .…”

Section: Discussionmentioning

confidence: 99%

“…As stated in [21] and [11,Lemma 5], non-dominated tuples (p, s, k) can be removed in linear time in the number of tuples if the tuples are different and sorted by profit. This is the case because every tuple in D (k) is stored in an array sorted according to the corresponding ξ.…”

Section: Remark 25mentioning

confidence: 99%

A faster FPTAS for the Unbounded Knapsack Problem

Jansen

¹

,

Kraft

²

2018

European Journal of Combinatorics

Self Cite

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The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0-1 Knapsack Problem (0-1 KP). In contrast to 0-1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time approximation schemes (FPTAS) are of great interest. Such algorithms find a solution arbitrarily close to the optimum OPT(I), i.e. of value at least (1 − ε)OPT(I) for ε > 0, and have a running time polynomial in the input length and

show abstract