[Paper Review] Genetic Algorithm for a class of Knapsack Problems

Shalin Shah|arXiv (Cornell University)|Feb 15, 2019

Optimization and Packing Problems5 references3 citations

TL;DR

This paper proposes a genetic algorithm tailored for spanner knapsack instances— a challenging class of 0/1 knapsack problems that resist traditional dynamic programming and core branch-and-bound methods due to high variable correlation. The algorithm achieves optimal solutions within reasonable computational time, demonstrating effectiveness on instances where prior exact methods fail.

ABSTRACT

The 0/1 knapsack problem is weakly NP-hard in that there exist pseudo-polynomial time algorithms based on dynamic programming that can solve it exactly. There are also the core branch and bound algorithms that can solve large randomly generated instances in a very short amount of time. However, as the correlation between the variables is increased, the difficulty of the problem increases. Recently a new class of knapsack problems was introduced by D. Pisinger called the spanner knapsack instances. These instances are unsolvable by the core branch and bound instances; and as the size of the coefficients and the capacity constraint increase, the spanner instances are unsolvable even by dynamic programming based algorithms. In this paper, a genetic algorithm is presented for spanner knapsack instances. Results show that the algorithm is capable of delivering optimum solutions within a reasonable amount of computational duration.

Motivation & Objective

• To address the computational intractability of spanner knapsack instances, which are resistant to standard exact algorithms.
• To develop a metaheuristic approach capable of solving large-scale, highly correlated knapsack problems.
• To evaluate the performance of a genetic algorithm on these difficult instances in terms of solution quality and runtime.

Proposed method

• A custom genetic algorithm is designed with binary encoding to represent knapsack solutions.
• Fitness evaluation is based on maximizing the total profit while respecting the capacity constraint.
• Selection uses tournament selection to promote high-fitness individuals.
• Crossover employs a uniform crossover strategy to explore diverse solution combinations.
• Mutation applies bit-flip operations with a low probability to maintain diversity.
• The algorithm is iteratively run over generations, with elitism preserving the best solutions.

Experimental results

Research questions

• RQ1 Can a genetic algorithm effectively solve spanner knapsack instances that are unsolvable by dynamic programming and core branch-and-bound methods?
• RQ2 How does the performance of the proposed genetic algorithm compare to exact methods on large, highly correlated knapsack instances?
• RQ3 What is the computational efficiency of the genetic algorithm in terms of runtime and convergence to optimal solutions?

Key findings

• The genetic algorithm successfully finds optimal solutions for spanner knapsack instances that are infeasible for dynamic programming and core branch-and-bound approaches.
• The algorithm achieves optimal results within a reasonable computational duration, even as instance size and coefficient magnitude increase.
• The solution quality remains consistently high across multiple test instances, indicating robustness to problem structure.
• The use of elitism and controlled mutation helps maintain diversity while converging toward optimal solutions.