A new upper bound for the multiple knapsack problem

TL;DR

This paper introduces a new upper bound for the Multiple Knapsack Problem by relaxing it to a divisible-size variant, demonstrating effectiveness especially when item-to-knapsack ratios are low.

Contribution

It proposes a novel relaxation technique called sequential relaxation, providing a tighter upper bound for MKP.

Findings

Effective for small item-to-knapsack ratios

Improves bounds on benchmark instances

Relaxation simplifies computation

Abstract

In this paper, a new upper bound for the Multiple Knapsack Problem (MKP) is proposed, based on the idea of relaxing MKP to a {\em Bounded Sequential Multiple Knapsack Problem}, i.e., a multiple knapsack problem in which item sizes are divisible. Such a relaxation, called sequential relaxation, is obtained by suitably replacing the items of a MKP instance with items with divisible sizes. Experimental results on benchmark instances show that the upper bound is effective when the ratio between the number of items and the number of knapsacks is small.