Near-Tight Approximation Algorithms for Bottleneck Multiple Knapsack Problems
This work provides the first near-optimal approximation algorithms for a classic combinatorial optimization problem, closing the gap between upper and lower bounds.
The paper presents near-tight approximation algorithms for the bottleneck multiple knapsack problem, achieving a (2/3 - ε)-approximation for identical capacities and a (1/2 - ε)-approximation for arbitrary capacities, matching known inapproximability bounds.
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 $\varepsilon > 0$.