A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem

Chekuri, Chandra; Khanna, Sanjeev

doi:10.1137/s0097539700382820

Cited by 336 publications

(307 citation statements)

References 27 publications

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“…There are several existing studies on the multiple knapsack problem. Chekuri and Khanna (2006) prove that the multiple knapsack problem does not admit an FPTAS even for the case with two knapsacks and give a polynomial time approximation scheme for the multiple knapsack problem. Dawande et al (2000) investigate the multiple knapsack problem with assignment restrictions where the objective is to maximize assigned weight and minimize utilized capacity.…”

Section: Related Literature and Problem Complexitymentioning

confidence: 99%

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

Tang

Meng

Chen

et al. 2016

M&SOM

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This paper investigates a practical batching decision problem that arises in the batch annealing operations in the cold rolling stage of steel production faced by most large iron and steel companies in the world. The problem is to select steel coils from a set of waiting coils to form batches to be annealed in available batch annealing furnaces and choose a median coil for each furnace. The objective is to maximize the total reward of the selected coils less the total coil-coil and coil-furnace mismatching cost. For a special case of the problem that arises frequently in practical settings where the coils are all similar and there is only one type of furnace available, we develop a polynomial-time dynamic programming algorithm to obtain an optimal solution. For the general case of the problem which is strongly NP-hard, an exact branch-and-price-and-cut solution algorithm is developed using a column and row generation framework. A variable reduction strategy is also proposed to accelerate the algorithm. The algorithm is capable of solving medium size instances to optimality within a reasonable computation time. In addition, a tabu search heuristic is proposed for solving larger instances. Three simple search neighborhoods as well as a sophisticated variable depth neighborhood are developed. This heuristic can generate near-optimal solutions for large instances within a short computation time.Using both randomly generated and real-world production data sets, it is shown that our algorithms are superior to a typical rule-based planning approach used by many steel plants. A decision support system that embeds our algorithms was developed and implemented at Baosteel to replace their rule-based planning method. The use of the system brings significant benefits to Baosteel, including an annual net profit increase of at least 1.76 million US Dollars, and a large reduction of standard coal consumption and carbon dioxide emissions.

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Section: Related Literature and Problem Complexitymentioning

confidence: 99%

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

Tang

Meng

Chen

et al. 2016

M&SOM

View full text Add to dashboard Cite

show abstract

“…There is also an option to choose which jobs to assign and which not. In [7] multiple results on MKP were presented. Using the result for GAP from [31], a 2-Approximation algorithm was provided.…”

Section: Allocation and Packing Problemsmentioning

confidence: 99%

Allocation algorithms for personal TV advertisements

2012

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In this paper we consider the problem of allocating personal TV advertisements to viewers. The problem's input consists of ad requests and viewers. Each ad is associated with a length, a payment, a requested number of viewers, a requested number of allocations per viewer and a target population profile. Each viewer is associated with a profile and an estimated viewing capacity which is uncertain. The goal is to maximize the revenue obtained from the allocation of ads to viewers for multiple periods while satisfying the ad constraints. First, we present the integer programming (IP) models of the problem and several heuristics for the deterministic version of the problem where the viewers' viewing capacities are known in advance. We compare the performances of the proposed algorithms to those of the state-of-the-art IP solver. Later, we discuss the multi-period uncertain problem and, based on the best heuristic for the deterministic version, present heuristics for low and high uncertainty. Through computational experiments, we evaluate our heuristics. For the deterministic version, our best heuristic attains 98 % of the possible revenue and for the multi-period uncertain version our heuristics performances are very high, even in cases of high uncertainty, compared to the revenue obtained by the deterministic version.

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“…We reduce a restricted version of the Generalized Assignment Problem (GAP) by Chekuri and Khanna [10] to the Integer Max-Flow WSNC problem with two power settings.…”

Section: Theorem 15mentioning

confidence: 99%

“…Chekuri and Khanna [10] show that the 2GAP-3 problem is APX-hard, hence it does not have a PTAS (unless P=NP).…”

Section: Theorem 15mentioning

confidence: 99%

Integer Maximum Flow in Wireless Sensor Networks with Energy Constraint

Bodlaender

Tan

Dijk

et al. 2008

Algorithm Theory – SWAT 2008

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We study the integer maximum flow problem on wireless sensor networks with energy constraint. In this problem, sensor nodes gather data and then relay them to a base station, before they run out of battery power. Packets are considered as integral units and not splittable. The problem is to find the maximum data flow in the sensor network subject to the energy constraint of the sensors. We show that this integral version of the problem is strongly NP-complete and in fact APX-hard. It follows that the problem is unlikely to have a polynomial time approximation scheme. Even when restricted to graphs with concrete geometrically defined connectivity and transmission costs, the problem is still strongly NP-complete. We provide some interesting polynomial time algorithms that give good approximations for the general case nonetheless. For networks with bounded treewidth greater than two, we show that the problem is weakly NP-complete and provide pseudo-polynomial time algorithms. For a special case of graphs with treewidth two, we give a polynomial time algorithm.

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A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem

Cited by 336 publications

References 27 publications

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

Allocation algorithms for personal TV advertisements

Integer Maximum Flow in Wireless Sensor Networks with Energy Constraint

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