Near-Tight Approximation Algorithms for Bottleneck Multiple Knapsack Problems

TL;DR

This paper develops near-tight approximation algorithms for the bottleneck multiple knapsack problem, achieving bounds close to known inapproximability limits for both identical and arbitrary capacities.

Contribution

It introduces approximation algorithms with bounds close to the theoretical limits for the bottleneck multiple knapsack problem, advancing the understanding of its computational complexity.

Findings

$(2/3 - ext{ε})$-approximation for identical capacities

$(1/2 - ext{ε})$-approximation for arbitrary capacities

Proves hardness bounds close to the approximation ratios

Abstract

In the bottleneck multiple knapsack problem, we are given a set of items and a set of knapsacks, where each item has a profit and a weight, and each knapsack has a capacity. Our goal is to assign items to knapsacks so as to maximize the minimum profit received by any knapsack subject to the capacity constraint. When all knapsacks have identical capacity, we give a $(\frac{2}{3} - \varepsilon)$-approximation algorithm for any constant $\varepsilon > 0$. This result almost matches the $(\frac{2}{3} + \varepsilon)$ inapproximability bound for the bottleneck multiple subset sum problem (Caprara et al., 2000). When the knapsacks can have arbitrary capacities, we propose a $(\frac{1}{2} - \varepsilon)$-approximation algorithm for any constant $\varepsilon > 0$. We also prove a hardness bound of $(\frac{1}{2} + \varepsilon)$ for any constant $\varepsilon > 0$.