Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size

The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence. Against this background, we study the expressive power of neural networks through the example of the classical NP-hard Knapsack Problem. Our main contribution is a class of recurrent neural networks (RNNs) with rectified linear units that are iteratively applied to each item of a Knapsack instance and thereby compute optimal or provably good solution values. We show that an RNN of depth four and width depending quadratically on the profit of an optimum Knapsack solution is sufficient to find optimum Knapsack solutions. We also prove the following tradeoff between the size of an RNN and the quality of the computed Knapsack solution: for Knapsack instances consisting of $n$ items, an RNN of depth five and width $w$ computes a solution of value at least $1-\mathcal{O}(n^2/\sqrt{w})$ times the optimum solution value. Our results build upon a classical dynamic programming formulation of the Knapsack Problem as well as a careful rounding of profit values that are also at the core of the well-known fully polynomial-time approximation scheme for the Knapsack Problem. A carefully conducted computational study qualitatively supports our theoretical size bounds. Finally, we point out that our results can be generalized to many other combinatorial optimization problems that admit dynamic programming solution methods, such as various Shortest Path Problems, the Longest Common Subsequence Problem, and the Traveling Salesperson Problem.Comment: A short version of this paper appears in the proceedings of AAAI 202

Improvement of the total mass and operating time of Knapsack sprayer to Propel Cart Sprayer (PCS)

There are two types of background of the farmers which are the large scale and small scale of agriculture. Usually, the large scale farmers will use the motorize Knapsack Sprayer while the small scale farmers will use manual-operated Knapsack Sprayer. The motorize Knapsack Sprayer that uses by the large scale of agriculture farmers’ area is to save the cost and time [1]. Unfortunately, both types of Knapsack Sprayer have their own ineffectiveness and risk especially the manual Knapsack Sprayer. The farmers that use the manual Knapsack Sprayer will have to carry the heavy load at their back while spraying the pesticide. These are a very burden to the farmers, especially for the old farmers. The weight of the mixture carried can be up to 17 kilograms depends on the density of the mixture whereas the safe weight lifting legalize by OSHA is 22.68 kilograms which the load almost near to its limit for average man and will affect the body locomotion and bones structure is carried in a long term period [2]. The total sprayed area per full tank is 44.09 meters square. The process of spraying the pesticide will slow down because the farmers have to bring the heavy load

A Novel Genetic Algorithm using Helper Objectives for the 0-1 Knapsack Problem

The 0-1 knapsack problem is a well-known combinatorial optimisation problem. Approximation algorithms have been designed for solving it and they return provably good solutions within polynomial time. On the other hand, genetic algorithms are well suited for solving the knapsack problem and they find reasonably good solutions quickly. A naturally arising question is whether genetic algorithms are able to find solutions as good as approximation algorithms do. This paper presents a novel multi-objective optimisation genetic algorithm for solving the 0-1 knapsack problem. Experiment results show that the new algorithm outperforms its rivals, the greedy algorithm, mixed strategy genetic algorithm, and greedy algorithm + mixed strategy genetic algorithm