# Analysis of Divide & Conquer strategies for the 0-1 Minimization Knapsack Problem

## TL;DR

This paper evaluates divide and conquer heuristics for large 0-1 Minimization Knapsack Problems, balancing reduced computational complexity with solution quality through theoretical guarantees and empirical testing.

## Contribution

It introduces and assesses new divide and conquer strategies, providing theoretical guarantees and empirical analysis to optimize their effectiveness.

## Key findings

- Strategies significantly reduce computational time.
- Empirical results identify optimal parameters for different data types.
- Theoretical bounds support the heuristic approach.

## Abstract

We introduce and asses several Divide \& Conquer heuristic strategies aimed to solve large instances of the 0-1 Minimization Knapsack Problem. The method subdivides a large problem in two smaller ones (or recursive iterations of the same principle), to lower down the global computational complexity of the original problem, at the expense of a moderate loss of quality in the solution. Theoretical mathematical results are presented in order to guarantee an algorithmically successful application of the method and to suggest the potential strategies for its implementation. In contrast, due to the lack of theoretical results, the solution's quality deterioration is measured empirically by means of Monte Carlo simulations for several types and values of the chosen strategies. Finally, introducing parameters of efficiency we suggest the best strategies depending on the data input.