Breeding Diverse Packings for the Knapsack Problem by Means of Diversity-Tailored Evolutionary Algorithms

TL;DR

This paper introduces a diversity-focused evolutionary algorithm for the knapsack problem that generates a set of high-quality, structurally diverse solutions, especially effective under limited evaluation budgets.

Contribution

It proposes a simple $(er)EA with entropy-based diversity measures and biased mutation/crossover, tailored for diverse solution generation in the knapsack problem.

Findings

Biased mutation and crossover improve diversity under limited evaluations.

Standard mutation operators perform slightly better in long-term scenarios.

The approach effectively balances solution quality and structural diversity.

Abstract

In practise, it is often desirable to provide the decision-maker with a rich set of diverse solutions of decent quality instead of just a single solution. In this paper we study evolutionary diversity optimization for the knapsack problem (KP). Our goal is to evolve a population of solutions that all have a profit of at least $(1-\varepsilon)\cdot OPT$, where OPT is the value of an optimal solution. Furthermore, they should differ in structure with respect to an entropy-based diversity measure. To this end we propose a simple $(\mu+1)$-EA with initial approximate solutions calculated by a well-known FPTAS for the KP. We investigate the effect of different standard mutation operators and introduce biased mutation and crossover which puts strong probability on flipping bits of low and/or high frequency within the population. An experimental study on different instances and settings shows that the proposed mutation operators in most cases perform slightly inferior in the long term, but show strong benefits if the number of function evaluations is severely limited.