Approximability of Two Variants of Multiple Knapsack Problems

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.