# Logical and Inequality Implications for Reducing the Size and Complexity of Quadratic Unconstrained Binary Optimization Problems

## TL;DR

This paper introduces new logical and inequality-based rules for preprocessing QUBO problems, significantly reducing their size and complexity, thereby enhancing solution efficiency and enabling larger problem instances to be tackled.

## Contribution

It extends previous reduction rules with additional ones, achieving exact solutions for 10% of benchmark QUBO problems and providing an efficient implementation algorithm.

## Key findings

- Successfully reduces QUBO problem sizes
- Achieves exact solutions for 10% of benchmarks
- Demonstrates efficacy through computational experiments

## Abstract

The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper (Lewis and Glover, 2016) a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.

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Source: https://tomesphere.com/paper/1705.09545