# Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions

@article{Dawande2000ApproximationAF, title={Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions}, author={Milind Dawande and Jayant Kalagnanam and Pınar Keskinocak and F. Sibel Salman and Ramamoorthi Ravi}, journal={Journal of Combinatorial Optimization}, year={2000}, volume={4}, pages={171-186} }

Motivated by a real world application, we study the multiple knapsack problem with assignment restrictions (MKAR). We are given a set of items, each with a positive real weight, and a set of knapsacks, each with a positive real capacity. In addition, for each item a set of knapsacks that can hold that item is specified. In a feasible assignment of items to knapsacks, each item is assigned to at most one knapsack, assignment restrictions are satisfied, and knapsack capacities are not exceeded…

## 133 Citations

Approximation Algorithms for the Generalized Multiple Knapsack Problems with K Restricted Elements

- Computer Science2015 7th International Conference on Intelligent Human-Machine Systems and Cybernetics
- 2015

The two problems are NP-complete when k is greater than or equal to 4, and the 1/2-approximation algorithm can be obtained and especially when k=2, this algorithm is an optimal algorithm.

A PTAS for the multiple knapsack problem

- Computer ScienceSODA '00
- 2000

The main result of this paper is a polynomial time approximation scheme for MKP, which helps demarcate the boundary at which instances of GAP become APX-hard.

Upper and lower bounding procedures for the multiple knapsack assignment problem

- Computer ScienceEur. J. Oper. Res.
- 2014

A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem

- Computer ScienceSIAM J. Comput.
- 2005

A polynomial time approximation scheme (PTAS) for MKP, which appears to be the strongest special case of GAP that is not APX-hard, and a PTAS-preserving reduction from an arbitrary instance of MKP to an instance with distinct sizes and profits.

Approximability of Two Variants of Multiple Knapsack Problems

- Computer ScienceCIAC
- 2015

A polynomial-time approximation algorithm for the MK-AR-CC and a lower bound on the approximation ratio of the algorithm is given, which shows that approximating the S-MK-AR with the ratio of $$n^{1-\epsilon }$$ is NP-hard even when all the items have the same profit.

Bi-Criteria Multiple Knapsack Problem with Grouped Items

- Computer Science, BusinessJ. Heuristics
- 2021

Algorithms which guarantee that rewards are not less than the optimal solution, with a bound on exceeded knapsack capacities are proposed, and a binary-search heuristic combined with these algorithms are proposed to obtain capacity-feasible solutions.

An approximation algorithm for a generalized assignment problem with small resource requirements

- Computer Science, Mathematics
- 2003

It is proved that the LP-relaxation of a special case of the generalized assignment problem is half-integral, and a weak persistency property is derived, and this approach leads to a ~-approximation algorithm.

DP-BASED ALGORITHM AND FPTAS FOR THE KNAPSACK SHARING AND RELATED PROBLEMS

- Computer ScienceJournal of the Operations Research Society of Japan
- 2019

A dynamic programmingbased (DP-based), pseudo-polynomial time algorithm is presented to solve XKSP to optimality in a unified way and an algorithm to solve the problem approximately in polynomial time by decomposing it into a series of subproblems is developed.

Multiple subset sum with inclusive assignment set restrictions

- Business, Mathematics
- 2011

In a traditional multiple subset sum problem (MSSP), there is a given set of items and a given set of bins (or knapsacks) with identical capacities. The objective is to select a subset of the items…

A 3/4-Approximation Algorithm for Multiple Subset Sum

- Computer ScienceJ. Heuristics
- 2003

A polynomial-time 3/4-approximation algorithm which runs fast also in practice and gets the approximation guarantee for a natural greedy heuristic for the 3-Partitioning Problem.

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