CIAC 2015: Algorithms and Complexity pp 365-376

# Approximability of Two Variants of Multiple Knapsack Problems

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)

## Abstract

This paper considers two variants of Multiple Knapsack Problems. The first one is the Multiple Knapsack Problem with Assignment Restrictions and Capacity Constraints (MK-AR-CC). In the MK-AR-CC($$k$$) (where $$k$$ is a positive integer), a subset of knapsacks is associated with each item and the item can be packed into only those knapsacks (Assignment Restrictions). Furthermore, the size of each knapsack is at least $$k$$ times the largest item assignable to the knapsack (Capacity Constraints). The MK-AR-CC($$k$$) is NP-hard for any constant $$k$$. In this paper, we give a polynomial-time $$\left( 1+\frac{2}{k+1}+\epsilon \right)$$-approximation algorithm for the MK-AR-CC($$k$$), and give a lower bound on the approximation ratio of our algorithm by showing an integrality gap of $$\left( 1+\frac{1}{k}-\epsilon \right)$$ for the IP formulation we use in our algorithm, where $$\epsilon$$ is an arbitrary small positive constant. The second problem is the Splittable Multiple Knapsack Problem with Assignment Restrictions (S-MK-AR), in which the size of items may exceed the capacity of knapsacks and items can be split and packed into multiple knapsacks. We show 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, where $$n$$ is the number of items and $$\epsilon$$ is an arbitrary positive constant.

## Keywords

Multiple knapsack problem Assignment restrictions Approximation algorithms

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Shuichi Miyazaki
• 1
• Naoyuki Morimoto
• 2
• Yasuo Okabe
• 1
1. 1.Academic Center for Computing and Media StudiesKyoto UniversityKyotoJapan
2. 2.Institute for Integrated Cell-Material Sciences (iCeMS)Kyoto UniversitySakyo-kuJapan